Fractal electrodynamics via non-integer dimensional space approach
Energy Technology Data Exchange (ETDEWEB)
Tarasov, Vasily E., E-mail: tarasov@theory.sinp.msu.ru
2015-09-25
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. - Highlights: • Electrodynamics of fractal media is described by non-integer dimensional spaces. • Applications of the fractal Gauss's and Ampere's laws are suggested. • Fractal Poisson equation, equation for fractal stream of charges are considered.
Anisotropic fractal media by vector calculus in non-integer dimensional space
Energy Technology Data Exchange (ETDEWEB)
Tarasov, Vasily E., E-mail: tarasov@theory.sinp.msu.ru [Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow 119991 (Russian Federation)
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 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.
The Cosmic Microwave Background Spectrum and a Determination of Fractal Space Dimensionality
Caruso, Francisco
2009-01-01
The possibility to constrain fractal space dimensionality form Astrophysics and other areas is briefly reviewed. Using data from FIRAS instrument aboard COBE satellite and assuming space dimensionality to be $3 + \\epsilon$, we calculate $\\epsilon = - (0.957 \\pm 0.006) \\times 10^{-5}$ and an absolute temperature 2.726 $\\pm$ 0.00003 K by fitting the cosmic microwave background radiation spectrum to Planck's radiation distribution.
A simpler and elegant algorithm for computing fractal dimension in higher dimensional state space
Indian Academy of Sciences (India)
S Ghorui; A K Das; N Venkatramani
2000-02-01
Chaotic systems are now frequently encountered in almost all branches of sciences. Dimension of such systems provides an important measure for easy characterization of dynamics of the systems. Conventional algorithms for computing dimension of such systems in higher dimensional state space face an unavoidable problem of enormous storage requirement. Here we present an algorithm, which uses a simple but very powerful technique and faces no problem in computing dimension in higher dimensional state space. The unique indexing technique of hypercubes, used in this algorithm, provides a clever means to drastically reduce the requirement of storage. It is shown that theoretically this algorithm faces no problem in computing capacity dimension in any dimension of the embedding state space as far as the actual dimension of the attractor is ﬁnite. Unlike the existing algorithms, memory requirement offered by this algorithm depends only on the actual dimension of the attractor and has no explicit dependence on the number of data points considered.
Fractal zeta functions and fractal drums higher-dimensional theory of complex dimensions
Lapidus, Michel L; Žubrinić, Darko
2017-01-01
This monograph gives a state-of-the-art and accessible treatment of a new general higher-dimensional theory of complex dimensions, valid for arbitrary bounded subsets of Euclidean spaces, as well as for their natural generalization, relative fractal drums. It provides a significant extension of the existing theory of zeta functions for fractal strings to fractal sets and arbitrary bounded sets in Euclidean spaces of any dimension. Two new classes of fractal zeta functions are introduced, namely, the distance and tube zeta functions of bounded sets, and their key properties are investigated. The theory is developed step-by-step at a slow pace, and every step is well motivated by numerous examples, historical remarks and comments, relating the objects under investigation to other concepts. Special emphasis is placed on the study of complex dimensions of bounded sets and their connections with the notions of Minkowski content and Minkowski measurability, as well as on fractal tube formulas. It is shown for the f...
Fractals and spectra related to fourier analysis and function spaces
Triebel, Hans
1997-01-01
Fractals and Spectra Hans Triebel This book deals with the symbiotic relationship between the theory of function spaces, fractal geometry, and spectral theory of (fractal) pseudodifferential operators as it has emerged quite recently. Atomic and quarkonial (subatomic) decompositions in scalar and vector valued function spaces on the euclidean n-space pave the way to study properties (compact embeddings, entropy numbers) of function spaces on and of fractals. On this basis, distributions of eigenvalues of fractal (pseudo)differential operators are investigated. Diverse versions of fractal drums are played. The book is directed to mathematicians interested in functional analysis, the theory of function spaces, fractal geometry, partial and pseudodifferential operators, and, in particular, in how these domains are interrelated. ------ It is worth mentioning that there is virtually no literature on this topic and hence the most of the presented material is published here the first time. - Zentralblatt MATH (…) ...
Searching for fractal phenomena in multidimensional phase-spaces
Blažek, Mikuláš
2000-07-01
A unified point of view on the fractal analysis in d-dimensional phase-spaces is presented. It is applicable to the data coming from the counting experiments. Explicit expressions are formulated for the fundamental types of factorial moments characterizing the presence of the fractal phenomena, their number being given by (2 d+1 - 1), as well as for a variety of associated statistical moments; special attention is paid to two and three dimensions. In particular, it is found that scaling properties of the modified dispersion moments are directly related with the presence of empty bins in the corresponding distributions. As to the high-energy experiments, those expressions can be applied to the data presently available, e.g. from LEP, as well as to the data arising in the near future from heavy-ion collisions performed at the CERN collider and from the pp collisions observed at the Tevatron, Fermilab.
Three-dimensional tumor perfusion reconstruction using fractal interpolation functions.
Craciunescu, O I; Das, S K; Poulson, J M; Samulski, T V
2001-04-01
It has been shown that the perfusion of blood in tumor tissue can be approximated using the relative perfusion index determined from dynamic contrast-enhanced magnetic resonance imaging (DE-MRI) of the tumor blood pool. Also, it was concluded in a previous report that the blood perfusion in a two-dimensional (2-D) tumor vessel network has a fractal structure and that the evolution of the perfusion front can be characterized using invasion percolation. In this paper, the three-dimensional (3-D) tumor perfusion is reconstructed from the 2-D slices using the method of fractal interpolation functions (FIF), i.e., the piecewise self-affine fractal interpolation model (PSAFIM) and the piecewise hidden variable fractal interpolation model (PHVFIM). The fractal models are compared to classical interpolation techniques (linear, spline, polynomial) by means of determining the 2-D fractal dimension of the reconstructed slices. Using FIFs instead of classical interpolation techniques better conserves the fractal-like structure of the perfusion data. Among the two FIF methods, PHVFIM conserves the 3-D fractality better due to the cross correlation that exists between the data in the 2-D slices and the data along the reconstructed direction. The 3-D structures resulting from PHVFIM have a fractal dimension within 3%-5% of the one reported in literature for 3-D percolation. It is, thus, concluded that the reconstructed 3-D perfusion has a percolation-like scaling. As the perfusion term from bio-heat equation is possibly better described by reconstruction via fractal interpolation, a more suitable computation of the temperature field induced during hyperthermia treatments is expected.
Wetting characteristics of 3-dimensional nanostructured fractal surfaces
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.
[Dimensional fractal of post-paddy wheat root architecture].
Chen, Xin-xin; Ding, Qi-shuo; Li, Yi-nian; Xue, Jin-lin; Lu, Ming-zhou; Qiu, Wei
2015-06-01
To evaluate whether crop rooting system was directionally dependent, a field digitizer was used to measure post-paddy wheat root architectures. The acquired data was transferred to Pro-E, in which virtual root architecture was reconstructed and projected to a series of planes each separated in 10° apart. Fractal dimension and fractal abundance of root projections in all the 18 planes were calculated, revealing a distinctive architectural distribution of wheat root in each direction. This strongly proved that post-paddy wheat root architecture was directionally dependent. From seedling to turning green stage, fractal dimension of the 18 projections fluctuated significantly, illustrating a dynamical root developing process in the period. At the jointing stage, however, fractal indices of wheat root architecture resumed its regularity in each dimension. This wheat root architecture recovered its dimensional distinctness. The proposed method was applicable for precision modeling field state root distribution in soil.
Electromagnetic space-time crystals. II. Fractal computational approach
Borzdov, G. N.
2014-01-01
A fractal approach to numerical analysis of electromagnetic space-time crystals, created by three standing plane harmonic waves with mutually orthogonal phase planes and the same frequency, is presented. Finite models of electromagnetic crystals are introduced, which make possible to obtain various approximate solutions of the Dirac equation. A criterion for evaluating accuracy of these approximate solutions is suggested.
Electromagnetic space-time crystals. II. Fractal computational approach
2014-01-01
A fractal approach to numerical analysis of electromagnetic space-time crystals, created by three standing plane harmonic waves with mutually orthogonal phase planes and the same frequency, is presented. Finite models of electromagnetic crystals are introduced, which make possible to obtain various approximate solutions of the Dirac equation. A criterion for evaluating accuracy of these approximate solutions is suggested.
Directory of Open Access Journals (Sweden)
Le Wang
2015-11-01
Full Text Available Based on phase space reconstruction and fractal dynamics in nonlinear dynamics, a method is proposed to extract and analyze the dynamics of the rotating stall in the impeller of centrifugal compressor, and some numerical examples are given to verify the results as well. First, the rotating stall of an existing low speed centrifugal compressor (LSCC is numerically simulated, and the time series of pressure in the rotating stall is obtained at various locations near the impeller outlet. Then, the phase space reconstruction is applied to these pressure time series, and a low-dimensional dynamical system, which the dynamics properties are included in, is reconstructed. In phase space reconstruction, C–C method is used to obtain the key parameters, such as time delay and the embedding dimension of the reconstructed phase space. Further, the fractal characteristics of the rotating stall are analyzed in detail, and the fractal dimensions are given for some examples to measure the complexity of the flow in the post-rotating stall. The results show that the fractal structures could reveal the intrinsic dynamics of the rotating stall flow and could be considered as a characteristic to identify the rotating stall.
Bells Galore: Oscillations and circle-map dynamics from space-filling fractal functions
Energy Technology Data Exchange (ETDEWEB)
Puente, C.E.; Cortis, A.; Sivakumar, B.
2008-10-15
The construction of a host of interesting patterns over one and two dimensions, as transformations of multifractal measures via fractal interpolating functions related to simple affine mappings, is reviewed. It is illustrated that, while space-filling fractal functions most commonly yield limiting Gaussian distribution measures (bells), there are also situations (depending on the affine mappings parameters) in which there is no limit. Specifically, the one-dimensional case may result in oscillations between two bells, whereas the two-dimensional case may give rise to unexpected circle map dynamics of an arbitrary number of two-dimensional circular bells. It is also shown that, despite the multitude of bells over two dimensions, whose means dance making regular polygons or stars inscribed on a circle, the iteration of affine maps yields exotic kaleidoscopes that decompose such an oscillatory pattern in a way that is similar to the many cases that converge to a single bell.
Quantized Fractal Space Time and Stochastic Holism
Sidharth, B G
2000-01-01
The space time that is used in relativistic Quantum Mechanics and Quantum Field Theory is the Minkowski space time. Yet, as pointed out by several scholars this classical space time is incompatible with the Heisenberg Uncertainity Principle: We cannot go down to arbitrarily small space time intervals, let alone space time points. Infact this classical space time is at best an approximation, and this has been criticised by several scholars. We investigate, what exactly this approximation entails.
Scale relativity and fractal space-time: theory and applications
Nottale, Laurent
2008-01-01
In the first part of this contribution, we review the development of the theory of scale relativity and its geometric framework constructed in terms of a fractal and nondifferentiable continuous space-time. This theory leads (i) to a generalization of possible physically relevant fractal laws, written as partial differential equation acting in the space of scales, and (ii) to a new geometric foundation of quantum mechanics and gauge field theories and their possible generalisations. In the second part, we discuss some examples of application of the theory to various sciences, in particular in cases when the theoretical predictions have been validated by new or updated observational and experimental data. This includes predictions in physics and cosmology (value of the QCD coupling and of the cosmological constant), to astrophysics and gravitational structure formation (distances of extrasolar planets to their stars, of Kuiper belt objects, value of solar and solar-like star cycles), to sciences of life (log-p...
On fractal space-time and fractional calculus
Directory of Open Access Journals (Sweden)
Hu Yue
2016-01-01
Full Text Available This paper gives an explanation of fractional calculus in fractal space-time. On observable scales, continuum models can be used, however, when the scale tends to a smaller threshold, a fractional model has to be adopted to describe phenomena in micro/nano structure. A time-fractional Fornberg-Whitham equation is used as an example to elucidate the physical meaning of the fractional order, and its solution process is given by the fractional complex transform.
Scale-free networks embedded in fractal space
Yakubo, K.; Korošak, D.
2011-06-01
The impact of an inhomogeneous arrangement of nodes in space on a network organization cannot be neglected in most real-world scale-free networks. Here we propose a model for a geographical network with nodes embedded in a fractal space in which we can tune the network heterogeneity by varying the strength of the spatial embedding. When the nodes in such networks have power-law distributed intrinsic weights, the networks are scale-free with the degree distribution exponent decreasing with increasing fractal dimension if the spatial embedding is strong enough, while the weakly embedded networks are still scale-free but the degree exponent is equal to γ=2 regardless of the fractal dimension. We show that this phenomenon is related to the transition from a noncompact to compact phase of the network and that this transition accompanies a drastic change of the network efficiency. We test our analytically derived predictions on the real-world example of networks describing the soil porous architecture.
The Art of Space Filling in Penrose Tilings and Fractals
Le, San
2011-01-01
Incorporating designs into the tiles that form tessellations presents an interesting challenge for artists. Creating a viable MC Escher like image that works esthetically as well as functionally requires resolving incongruencies at a tile's edge while constrained by its shape. Escher was the most well known practitioner in this style of mathematical visualization, but there are significant mathematical shapes to which he never applied his artistry. These shapes can incorporate designs that form images as appealing as those produced by Escher, and our paper explores this for traditional tessellations, Penrose Tilings, fractals, and fractal/tessellation combinations. To illustrate the versatility of tiling art, images were created with multiple figures and negative space leading to patterns distinct from the work of others.
Chaos and Fractals in a (2+1)-Dimensional Soliton System
Institute of Scientific and Technical Information of China (English)
郑春龙; 张解放; 盛正卯
2003-01-01
Considering that there are abundant coherent soliton excitations in high dimensions, we reveal a novel phenomenon that the localized excitations possess chaotic and fractal behaviour in some (2+1)-dimensional soliton systems. To clarify the interesting phenomenon, we take the generalized (2+1)-dimensional Nizhnik-NovikovVesselov system as a concrete example. A quite general variable separation solutions of this system is derived via a variable separation approach first, then some new excitations like chaos and fractals are derived by introducing some types of lower-dimensional chaotic and fractal patterns.
Fractal spectra in generalized Fibonacci one-dimensional magnonic quasicrystals
Energy Technology Data Exchange (ETDEWEB)
Costa, C.H.O. [Departamento de Fisica Teorica e Experimental, Universidade Federal do Rio grande do Norte, 59072-970 Natal-RN (Brazil); Vasconcelos, M.S., E-mail: manoelvasconcelos@yahoo.com.br [Escola de Ciencias e Tecnologia, Universidade Federal do Rio grande do Norte, 59072-970 Natal-RN (Brazil); Barbosa, P.H.R.; Barbosa Filho, F.F. [Departamento de Fisica, Universidade Federal do Piaui, 64049-550 Teresina-Pi (Brazil)
2012-07-15
In this work we carry out a theoretical analysis of the spectra of magnons in quasiperiodic magnonic crystals arranged in accordance with generalized Fibonacci sequences in the exchange regime, by using a model based on a transfer-matrix method together random-phase approximation (RPA). The generalized Fibonacci sequences are characterized by an irrational parameter {sigma}(p,q), which rules the physical properties of the system. We discussed the magnonic fractal spectra for first three generalizations, i.e., silver, bronze and nickel mean. By varying the generation number, we have found that the fragmentation process of allowed bands makes possible the emergence of new allowed magnonic bulk bands in spectra regions that were magnonic band gaps before, such as which occurs in doped semiconductor devices. This interesting property arises in one-dimensional magnonic quasicrystals fabricated in accordance to quasiperiodic sequences, without the need to introduce some deferent atomic layer or defect in the system. We also make a qualitative and quantitative investigations on these magnonic spectra by analyzing the distribution and magnitude of allowed bulk bands in function of the generalized Fibonacci number F{sub n} and as well as how they scale as a function of the number of generations of the sequences, respectively. - Highlights: Black-Right-Pointing-Pointer Quasiperiodic magnonic crystals are arranged in accordance with the generalized Fibonacci sequence. Black-Right-Pointing-Pointer Heisenberg model in exchange regime is applied. Black-Right-Pointing-Pointer We use a theoretical model based on a transfer-matrix method together random-phase approximation. Black-Right-Pointing-Pointer Fractal spectra are characterized. Black-Right-Pointing-Pointer We analyze the distribution of allowed bulk bands in function of the generalized Fibonacci number.
Pulmonary vasculature in dogs assessed by three-dimensional fractal analysis and chemometrics
DEFF Research Database (Denmark)
Müller, Anna V; Marschner, Clara B; Kristensen, Annemarie T
2017-01-01
angiogram, applying fractal analyses of these vascular trees in dogs with and without diseases that are known to predispose to thromboembolism, and testing the hypothesis that diseased dogs would have a different fractal dimension than healthy dogs. A total of 34 dogs were sampled. Based on computed...... tomographic pulmonary angiograms findings, dogs were divided in three groups: diseased with pulmonary thromboembolism (n = 7), diseased but without pulmonary thromboembolism (n = 21), and healthy (n = 6). An observer who was aware of group status created three-dimensional pulmonary artery vascular trees...... for each dog using a semiautomated segmentation technique. Vascular three-dimensional reconstructions were then evaluated using fractal analysis. Fractal dimensions were analyzed, by group, using analysis of variance and principal component analysis. Fractal dimensions were significantly different among...
A novel schedule for solving the two-dimensional diffusion problem in fractal heat transfer
Directory of Open Access Journals (Sweden)
Xu Shu
2015-01-01
Full Text Available In this work, the local fractional variational iteration method is employed to obtain approximate analytical solution of the two-dimensional diffusion equation in fractal heat transfer with help of local fractional derivative and integral operators.
Electromagnetic Fields and Waves in Fractional Dimensional Space
Zubair, Muhammad; Naqvi, Qaisar Abbas
2012-01-01
This book presents the concept of fractional dimensional space applied to the use of electromagnetic fields and waves. It provides demonstrates the advantages in studying the behavior of electromagnetic fields and waves in fractal media. The book presents novel fractional space generalization of the differential electromagnetic equations is provided as well as a new form of vector differential operators is formulated in fractional space. Using these modified vector differential operators, the classical Maxwell's electromagnetic equations are worked out. The Laplace's, Poisson's and Helmholtz's
Chappard, D; Legrand, E; Haettich, B; Chalès, G; Auvinet, B; Eschard, J P; Hamelin, J P; Baslé, M F; Audran, M
2001-11-01
Trabecular bone has been reported as having two-dimensional (2-D) fractal characteristics at the histological level, a finding correlated with biomechanical properties. However, several fractal dimensions (D) are known and computational ways to obtain them vary considerably. This study compared three algorithms on the same series of bone biopsies, to obtain the Kolmogorov, Minkowski-Bouligand, and mass-radius fractal dimensions. The relationships with histomorphometric descriptors of the 2-D trabecular architecture were investigated. Bone biopsies were obtained from 148 osteoporotic male patients. Bone volume (BV/TV), trabecular characteristics (Tb.N, Tb.Sp, Tb.Th), strut analysis, star volumes (marrow spaces and trabeculae), inter-connectivity index, and Euler-Poincaré number were computed. The box-counting method was used to obtain the Kolmogorov dimension (D(k)), the dilatation method for the Minkowski-Bouligand dimension (D(MB)), and the sandbox for the mass-radius dimension (D(MR)) and lacunarity (L). Logarithmic relationships were observed between BV/TV and the fractal dimensions. The best correlation was obtained with D(MR) and the lowest with D(MB). Lacunarity was correlated with descriptors of the marrow cavities (ICI, star volume, Tb.Sp). Linear relationships were observed among the three fractal techniques which appeared highly correlated. A cluster analysis of all histomorphometric parameters provided a tree with three groups of descriptors: for trabeculae (Tb.Th, strut); for marrow cavities (Euler, ICI, Tb.Sp, star volume, L); and for the complexity of the network (Tb.N and the three D's). A sole fractal dimension cannot be used instead of the classic 2-D descriptors of architecture; D rather reflects the complexity of branching trabeculae. Computation time is also an important determinant when choosing one of these methods.
Two-Dimensional Fractal Metamaterials for Applications in THz
DEFF Research Database (Denmark)
Malureanu, Radu; Jepsen, Peter Uhd; Zalkovskij, Maksim
2011-01-01
applications. THz radiation can be employed for various purposes, among them the study of vibrations in biological molecules, motion of electrons in semiconductors and propagation of acoustic shock waves in crystals. We propose here a new THz fractal MTM design that shows very high transmission in the desired...... frequency range as well as a clear differentiation between one polarisation and another. Based on theoretical predictions we fabricated and measured a fractal based THz metamaterial that shows more than 60% field transmission at around 1 THz for TE polarized light while the TM waves have almost 80% field...
Steady laminar flow of fractal fluids
Balankin, Alexander S.; Mena, Baltasar; Susarrey, Orlando; Samayoa, Didier
2017-02-01
We study laminar flow of a fractal fluid in a cylindrical tube. A flow of the fractal fluid is mapped into a homogeneous flow in a fractional dimensional space with metric induced by the fractal topology. The equations of motion for an incompressible Stokes flow of the Newtonian fractal fluid are derived. It is found that the radial distribution for the velocity in a steady Poiseuille flow of a fractal fluid is governed by the fractal metric of the flow, whereas the pressure distribution along the flow direction depends on the fractal topology of flow, as well as on the fractal metric. The radial distribution of the fractal fluid velocity in a steady Couette flow between two concentric cylinders is also derived.
Institute of Scientific and Technical Information of China (English)
ZHENG Chun-Long; ZHU Jia-Min; ZHANG Jie-Fang; CHEN Li-Qun
2003-01-01
By means of variable separation approach, quite a general excitation of the new (2 + 1)-dimensional long dispersive wave system: λqt + qxx - 2q ∫ (qr)xdy ＝ 0, λrt - rxx + 2r ∫(qr)xdy ＝ 0, is derived. Some types of the usual localized excitations such as dromions, lumps, rings, and oscillating soliton excitations can be easily constructed by selecting the arbitrary functions appropriately. Besides these usual localized structures, some new localized excitations like fractal-dromion, farctal-lump, and multi-peakon excitations of this new system are found by selecting appropriate functions.
Changes in Dimensionality and Fractal Scaling Suggest Soft-Assembled Dynamics in Human EEG
DEFF Research Database (Denmark)
Wiltshire, Travis; Euler, Matthew J.; McKinney, Ty
2017-01-01
. In a repetition priming task, we assessed evidence for changes in the correlation dimension and fractal scaling exponents during stimulus-locked event-related potentials, as a function of stimulus onset and familiarity, and relative to spontaneous non-task-related activity. Consistent with predictions derived...... from soft-assembly, results indicated decreases in dimensionality and increases in fractal scaling exponents from resting to pre-stimulus states and following stimulus onset. However, contrary to predictions, familiarity tended to increase dimensionality estimates. Overall, the findings support...
Raikov, A. A.; Orlov, V. V.; Gerasim, R. V.
2014-06-01
A pairwise distance technique developed by the authors is used to identify signs of fractal structure in sets of extragalactic supernovae (822 type Ia supernovae in the regions 300° ≤ α ≤ 360° and 0° ≤ α ≤ 60° , - 5° ≤ δ ≤ 5°). Since the region of space occupied by the objects in the sample is highly oblate, we use Mandelbrot's codimensionality theorem. Three cosmological models are examined: a model with a euclidean metric, a "tired light" model, and the standard ΛCDM model. Estimates of a fractal dimensionality of D ≅ 2.69 are obtained for the first two models and D ≅ 2.64 for the ΛCDM model.
Polarimetric Wavelet Fractal Remote Sensing Principles for Space Materials (Preprint)
2012-06-04
introduced by Mandelbrot 15. Fractal geometry is the geometry of self-similarity in which an object appears to look similar at different scales. The key... Mandelbrot , The Fractal Geometry of Nature, W.H. Freeman and Co., New York, 1977. [16] I.M. Johnstone and B.W. Silverman, “Wavelet threshold estimators
Peakon Excitations and Fractal Dromions for General (2+1)-Dimensional Korteweg de Vries System
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
By means of an extended mapping approach and a linear variable separation approach, a new family of exact solutions of the general (2+1)-dimensional Korteweg de Vries system (GKdV) are derived. Based on the derived solitary wave excitation, we obtain some special peakon excitations and fractal dromions in this short note.
Ndiaye, Mambaye; Terranova, Lisa; Mallet, Romain; Mabilleau, Guillaume; Chappard, Daniel
2015-01-01
The macrophysical properties of granular biomaterials used to fill bone defects have rarely been considered. Granules of a given biomaterial occupy three-dimensional (3-D) space when packed together and create a macroporosity suitable for the invasion of vascular and bone cells. Granules of β-tricalcium phosphate were prepared using polyurethane foam technology and increasing the amount of material powder in the slurry (10, 11, 15, 18, 21 and 25 g). After sintering, granules of 1000-2000 μm were prepared by sieving. They were analyzed morphologically by scanning electron microscopy and placed in polyethylene test tubes to produce 3-D scaffolds. Microcomputed tomography (microCT) was used to image the scaffolds and to determine porosity and fractal dimension in three dimensions. Two-dimensional sections of the microCT models were binarized and used to compute classical morphometric parameters describing porosity (interconnectivity index, strut analysis and star volumes) and fractal dimensions. In addition, two newly important fractal parameters (lacunarity and succolarity) were measured. Compression analysis of the stacks of granules was done. Porosity decreased as the amount of material in the slurry increased but non-linear relationships were observed between microarchitectural parameters describing the pores and porosity. Lacunarity increased in the series of granules but succolarity (reflecting the penetration of a fluid) was maximal in the 15-18 g groups and decreased noticeably in the 25 g group. The 3-D arrangement of biomaterial granules studied by these new fractal techniques allows the optimal formulation to be derived based on the lowest amount of material, suitable mechanical resistance during crushing and the creation of large interconnected pores. Copyright © 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
DEFF Research Database (Denmark)
Bruun Jensen, Casper
2007-01-01
. Instead, I outline a fractal approach to the study of space, society, and infrastructure. A fractal orientation requires a number of related conceptual reorientations. It has implications for thinking about scale and perspective, and (sociotechnical) relations, and for considering the role of the social...... and a fractal social theory....
Understanding high-dimensional spaces
Skillicorn, David B
2012-01-01
High-dimensional spaces arise as a way of modelling datasets with many attributes. Such a dataset can be directly represented in a space spanned by its attributes, with each record represented as a point in the space with its position depending on its attribute values. Such spaces are not easy to work with because of their high dimensionality: our intuition about space is not reliable, and measures such as distance do not provide as clear information as we might expect. There are three main areas where complex high dimensionality and large datasets arise naturally: data collected by online ret
Institute of Scientific and Technical Information of China (English)
YANG Ming-yang; ZHOU Jun; L Petti; S De Nicola; P Mormile
2011-01-01
We report a numerical method to analyze the fractal characteristics of far-field diffraction patterns for two-dimensional Thue-Morse(2-D TM) structures.The far-field diffraction patterns of the 2-D TM structures can be obtained by the numerical method,and they have a good agreement with the experimental ones.The analysis shows that the fractal characteristics of far-field diffraction patterns for the 2-D TM structures are determined by the inflation rule,which have potential applications in the design of optical diffraction devices.
Two-Dimensional Fractal Metamaterials for Applications in THz
DEFF Research Database (Denmark)
Malureanu, Radu; Jepsen, Peter Uhd; Zalkovskij, Maksim
2011-01-01
The concept of metamaterials (MTMs) is acknowledged for providing new horizons for controlling electromagnetic radiations thus their use in frequency ranges otherwise difficult to manage (e.g. THz radiation) broadens our possibility to better understand our world as well as opens the path for new...... frequency range as well as a clear differentiation between one polarisation and another. Based on theoretical predictions we fabricated and measured a fractal based THz metamaterial that shows more than 60% field transmission at around 1 THz for TE polarized light while the TM waves have almost 80% field...
Fractal space-time fluctuations: A signature of quantumlike chaos in dynamical systems
Selvam, A M
2004-01-01
Dynamical systems in nature such as fluid flows, heart beat patterns, rainfall variability, stock market price fluctuations, etc. exhibit selfsimilar fractal fluctuations on all scales in space and time. Power spectral analyses of fractal fluctuations exhibit inverse power law form indicating long-range space-time correlations, identified as self-organized criticality. The author has proposed a general systems theory, which predicts the observed self-organized criticality as signatures of quantumlike chaos. The model shows that (1) the fractal fluctuations result from an overall logarithmic spiral trajectory with the quasiperiodic Penrose tiling pattern for the internal structure. Conventional power spectral analysis of such a logarithmic spiral trajectory will show a continuum of eddies with progressive increase in phase. (2) Power spectral analyses of fractal fluctuations of dynamical systems exhibit the universal inverse power law form of the statistical normal distribution. Such a result indicates that th...
Magnetic reconnection rate in space plasmas: a fractal approach.
Materassi, Massimo; Consolini, Giuseppe
2007-10-26
Magnetic reconnection is generally discussed via a fluid description. Here, we evaluate the reconnection rate assuming a fractal topology of the reconnection region. The central idea is that the fluid hypothesis may be violated at the scales where reconnection takes place. The reconnection rate, expressed as the Alfvén Mach number of the plasma moving toward the diffusion region, is shown to depend on the fractal dimension and on the sizes of the reconnection or diffusion region. This mechanism is more efficient than prediction of the Sweet-Parker model and even Petschek's model for finite magnetic Reynolds number. A good agreement also with rates given by Hall MHD models is found. A discussion of the fractal assumption on the diffusion region in terms of current microstructures is proposed. The comparison with in-situ satellite observations suggests the reconnection region to be a filamentary domain.
Experimental Evidence of Dynamical Scaling in a Two-Dimensional Fractal Growth
Miyashita, Satoru; Saito, Yukio; Uwaha, Makio
1997-04-01
A dynamical scaling law of fractal aggregation is testedusing electrochemical deposition without an external electric field.Silver metal leaves grow on the edge of a Cu plate placed in a thin cell containing an AgNO3-water solution due to the difference in ionization tendency between Ag and Cu. We find that the tip height h(t) satisfies the dynamical scaling relationh(t)= c-1/(d-D_f) \\tilde{g}(tc2/(d-D_f)) with respect to the solute concentration cin the space dimension d=2 with the fractal dimension Df=1.71 of the diffusion-limited aggregation.
Tumor growth in the space-time with temporal fractal dimension
Energy Technology Data Exchange (ETDEWEB)
Molski, Marcin [Department of Theoretical Chemistry, Faculty of Chemistry, Adam Mickiewicz University of Poznan, ul. Grunwaldzka 6, PL 60-780 Poznan (Poland)], E-mail: marcin@rovib.amu.edu.pl; Konarski, Jerzy [Department of Theoretical Chemistry, Faculty of Chemistry, Adam Mickiewicz University of Poznan, ul. Grunwaldzka 6, PL 60-780 Poznan (Poland)
2008-05-15
An improvement of the Waliszewski and Konarski approach [Waliszewski P, Konarski J. The Gompertzian curve reveals fractal properties of tumor growth. Chaos, Solitons and Fractals 2003;16:665-74] to determination of the time-dependent temporal fractal dimension b{sub t}(t) and the scaling factor a{sub t}(t) for the tumor formation in the fractal space-time is presented. The analytical formulae describing the time-dependence of b{sub t}(t) and a{sub t}(t), which take into account appropriate boundary conditions for t {yields} 0 and t {yields} {infinity}, are derived. Their validity is tested on the experimental growth curve obtained by Laird for the Flexner-Jobling rat's tumor. A hypothesis is formulated that tumorigenesis has a lot in common with the neuronal differentiation and synapse formation. These processes are qualitatively described by the same Gompertz function of growth and take place in the fractal space-time whose mean temporal fractal dimension is lost during progression.
5-Dimensional Extended Space Model
Tsipenyuk, D. Yu.; Andreev, V. A.
2006-01-01
We put forward an idea that physical phenomena have to be treated in 5-dimensional space where the fifth coordinate is the interval S. Thus, we considered the (1+4) extended space G(T;X,Y,Z,S). In addition to Lorentz transformations (T;X), (T;Y), (T;Z) which are in (1+3)-dimensional Minkowski space, in the proposed (1+4)d extended space two other types of transformations exist in planes (T,S); (X,S), (Y,S), (Z,S) that converts massive particles into massless and vice versa. We also consider e...
Tao, Xie; Shang-Zhuo, Zhao; William, Perrie; He, Fang; Wen-Jin, Yu; Yi-Jun, He
2016-06-01
To study the electromagnetic backscattering from a one-dimensional drifting fractal sea surface, a fractal sea surface wave-current model is derived, based on the mechanism of wave-current interactions. The numerical results show the effect of the ocean current on the wave. Wave amplitude decreases, wavelength and kurtosis of wave height increase, spectrum intensity decreases and shifts towards lower frequencies when the current occurs parallel to the direction of the ocean wave. By comparison, wave amplitude increases, wavelength and kurtosis of wave height decrease, spectrum intensity increases and shifts towards higher frequencies if the current is in the opposite direction to the direction of ocean wave. The wave-current interaction effect of the ocean current is much stronger than that of the nonlinear wave-wave interaction. The kurtosis of the nonlinear fractal ocean surface is larger than that of linear fractal ocean surface. The effect of the current on skewness of the probability distribution function is negligible. Therefore, the ocean wave spectrum is notably changed by the surface current and the change should be detectable in the electromagnetic backscattering signal. Project supported by the National Natural Science Foundation of China (Grant No. 41276187), the Global Change Research Program of China (Grant No. 2015CB953901), the Priority Academic Development Program of Jiangsu Higher Education Institutions (PAPD), Program for the Innovation Research and Entrepreneurship Team in Jiangsu Province, China, the Canadian Program on Energy Research and Development, and the Canadian World Class Tanker Safety Service.
Cantorian Fractal Space-Time Fluctuations in Turbulent Fluid Flows and the Kinetic Theory of Gases
Selvam, A M
1999-01-01
Fluid flows such as gases or liquids exhibit space-time fluctuations on all scales extending down to molecular scales. Such broadband continuum fluctuations characterise all dynamical systems in nature and are identified as selfsimilar fractals in the newly emerging multidisciplinary science of nonlinear dynamics and chaos. A cell dynamical system model has been developed by the author to quantify the fractal space-time fluctuations of atmospheric flows. The earth's atmosphere consists of a mixture of gases and obeys the gas laws as formulated in the kinetic theory of gases developed on probabilistic assumptions in 1859 by the physicist James Clerk Maxwell. An alternative theory using the concept of fractals and chaos is applied in this paper to derive these fundamental gas laws.
Fractal Systems of Central Places Based on Intermittency of Space-filling
Chen, Yanguang
2011-01-01
The central place models are fundamentally important in theoretical geography and city planning theory. The texture and structure of central place networks have been demonstrated to be self-similar in both theoretical and empirical studies. However, the underlying rationale of central place fractals in the real world has not yet been revealed so far. This paper is devoted to illustrating the mechanisms by which the fractal patterns can be generated from central place systems. The structural dimension of the traditional central place models is d=2 indicating no intermittency in the spatial distribution of human settlements. This dimension value is inconsistent with empirical observations. Substituting the complete space filling with the incomplete space filling, we can obtain central place models with fractional dimension D
Steady laminar flow of fractal fluids
Energy Technology Data Exchange (ETDEWEB)
Balankin, Alexander S., E-mail: abalankin@ipn.mx [Grupo Mecánica Fractal, ESIME, Instituto Politécnico Nacional, México D.F., 07738 (Mexico); Mena, Baltasar [Laboratorio de Ingeniería y Procesos Costeros, Instituto de Ingeniería, Universidad Nacional Autónoma de México, Sisal, Yucatán, 97355 (Mexico); Susarrey, Orlando; Samayoa, Didier [Grupo Mecánica Fractal, ESIME, Instituto Politécnico Nacional, México D.F., 07738 (Mexico)
2017-02-12
We study laminar flow of a fractal fluid in a cylindrical tube. A flow of the fractal fluid is mapped into a homogeneous flow in a fractional dimensional space with metric induced by the fractal topology. The equations of motion for an incompressible Stokes flow of the Newtonian fractal fluid are derived. It is found that the radial distribution for the velocity in a steady Poiseuille flow of a fractal fluid is governed by the fractal metric of the flow, whereas the pressure distribution along the flow direction depends on the fractal topology of flow, as well as on the fractal metric. The radial distribution of the fractal fluid velocity in a steady Couette flow between two concentric cylinders is also derived. - Highlights: • Equations of Stokes flow of Newtonian fractal fluid are derived. • Pressure distribution in the Newtonian fractal fluid is derived. • Velocity distribution in Poiseuille flow of fractal fluid is found. • Velocity distribution in a steady Couette flow is established.
Numerical study for the c-dependence of fractal dimension in two-dimensional quantum gravity
Kawamoto, N; Kawamoto, Noboru; Yotsuji, Kenji
2002-01-01
We numerically investigate the fractal structure of two-dimensional quantum gravity coupled to matter central charge c for $-2 \\leq c \\leq 1$. We reformulate Q-state Potts model into the model which can be identified as a weighted percolation cluster model and can make continuous change of Q, which relates c, on the dynamically triangulated lattice. The c-dependence of the critical coupling is measured from the percolation probability and susceptibility. The c-dependence of the string susceptibility of the quantum surface is evaluated and has very good agreement with the theoretical predictions. The c-dependence of the fractal dimension based on the finite size scaling hypothesis is measured and has excellent agreement with one of the theoretical predictions previously proposed except for the region near $c\\approx 1$.
Mona Lisa:. the Stochastic View and Fractality in Color Space
Pedram, Pouria; Jafari, G. R.
A painting consists of objects which are arranged in specific ways. The art of painting is drawing the objects, which can be considered as known trends, in an expressive manner. Detrended methods are suitable for characterizing the artistic works of the painter by eliminating trends. It means that the study of paintings, regardless of its apparent purpose, as a stochastic process. Multifractal detrended fluctuation analysis is applied to characterize the statistical properties of Mona Lisa, as an instance, to exhibit the fractality of the painting. The results show that Mona Lisa is a long-range correlated and almost behaves similar in various scales.
Synthesis of fractal light pulses by quasi-direct space-to-time pulse shaping.
Mendoza-Yero, Omel; Alonso, Benjamín; Mínguez-Vega, Gladys; Sola, Iñigo Juan; Lancis, Jesús; Monsoriu, Juan A
2012-04-01
We demonstrated a simple diffractive method to map the self-similar structure shown in squared radial coordinate of any set of circularly symmetric fractal plates into self-similar light pulses in the corresponding temporal domain. The space-to-time mapping of the plates was carried out by means of a kinoform diffractive lens under femtosecond illumination. The spatio-temporal characteristics of the fractal pulses obtained in this way were measured by means of a spectral interferometry technique assisted by a fiber optics coupler (STARFISH). Our proposal allows synthesizing suited sequences of focused fractal femtosecond pulses potentially useful for several current applications, such as femtosecond material processing, atomic, and molecular control of chemical processes or generation of nonlinear effects.
Tao, Xie; William, Perrie; Shang-Zhuo, Zhao; He, Fang; Wen-Jin, Yu; Yi-Jun, He
2016-07-01
Sea surface current has a significant influence on electromagnetic (EM) backscattering signals and may constitute a dominant synthetic aperture radar (SAR) imaging mechanism. An effective EM backscattering model for a one-dimensional drifting fractal sea surface is presented in this paper. This model is used to simulate EM backscattering signals from the drifting sea surface. Numerical results show that ocean currents have a significant influence on EM backscattering signals from the sea surface. The normalized radar cross section (NRCS) discrepancies between the model for a coupled wave-current fractal sea surface and the model for an uncoupled fractal sea surface increase with the increase of incidence angle, as well as with increasing ocean currents. Ocean currents that are parallel to the direction of the wave can weaken the EM backscattering signal intensity, while the EM backscattering signal is intensified by ocean currents propagating oppositely to the wave direction. The model presented in this paper can be used to study the SAR imaging mechanism for a drifting sea surface. Project supported by the National Natural Science Foundation of China (Grant No. 41276187), the Global Change Research Program of China (Grant No. 2015CB953901), the Priority Academic Program Development of Jiangsu Higher Education Institutions, China, the Program for the Innovation Research and Entrepreneurship Team in Jiangsu Province, China, the Canadian Program on Energy Research and Development, and the Canadian World Class Tanker Safety Service Program.
Directory of Open Access Journals (Sweden)
Franceschini Barbara
2005-02-01
Full Text Available Abstract Background Modeling the complex development and growth of tumor angiogenesis using mathematics and biological data is a burgeoning area of cancer research. Architectural complexity is the main feature of every anatomical system, including organs, tissues, cells and sub-cellular entities. The vascular system is a complex network whose geometrical characteristics cannot be properly defined using the principles of Euclidean geometry, which is only capable of interpreting regular and smooth objects that are almost impossible to find in Nature. However, fractal geometry is a more powerful means of quantifying the spatial complexity of real objects. Methods This paper introduces the surface fractal dimension (Ds as a numerical index of the two-dimensional (2-D geometrical complexity of tumor vascular networks, and their behavior during computer-simulated changes in vessel density and distribution. Results We show that Ds significantly depends on the number of vessels and their pattern of distribution. This demonstrates that the quantitative evaluation of the 2-D geometrical complexity of tumor vascular systems can be useful not only to measure its complex architecture, but also to model its development and growth. Conclusions Studying the fractal properties of neovascularity induces reflections upon the real significance of the complex form of branched anatomical structures, in an attempt to define more appropriate methods of describing them quantitatively. This knowledge can be used to predict the aggressiveness of malignant tumors and design compounds that can halt the process of angiogenesis and influence tumor growth.
Institute of Scientific and Technical Information of China (English)
谢涛; William Perrie; 赵尚卓; 方贺; 于文金; 何宜军
2016-01-01
Sea surface current has a significant influence on electromagnetic (EM) backscattering signals and may constitute a dominant synthetic aperture radar (SAR) imaging mechanism. An effective EM backscattering model for a one-dimensional drifting fractal sea surface is presented in this paper. This model is used to simulate EM backscattering signals from the drifting sea surface. Numerical results show that ocean currents have a significant influence on EM backscattering signals from the sea surface. The normalized radar cross section (NRCS) discrepancies between the model for a coupled wave-current fractal sea surface and the model for an uncoupled fractal sea surface increase with the increase of incidence angle, as well as with increasing ocean currents. Ocean currents that are parallel to the direction of the wave can weaken the EM backscattering signal intensity, while the EM backscattering signal is intensified by ocean currents propagating oppositely to the wave direction. The model presented in this paper can be used to study the SAR imaging mechanism for a drifting sea surface.
Multi-Dimensional Piece-Wise Self-Affine Fractal Interpolation Model
Institute of Scientific and Technical Information of China (English)
ZHANG Tong; ZHUANG Zhuo
2007-01-01
Iterated function system (IFS) models have been used to represent discrete sequences where the attractor of the IFS is piece-wise self-affine in R2 or R3 (R is the set of real numbers). In this paper, the piece-wise self-affine IFS model is extended from R3 to Rn (n is an integer greater than 3), which is called the multi-dimensional piece-wise self-affine fractal interpolation model. This model uses a "mapping partial derivative", and a constrained inverse algorithm to identify the model parameters. The model values depend continuously on all the model parameters, and represent most data which are not multi-dimensional self-affine in Rn. Therefore, the result is very general. The class of functions obtained is much more diverse because their values depend continuously on all of the variables, with all the coefficients of the possible multi-dimensional affine maps determining the functions.
Neuronal differentiation and synapse formation occur in space and time with fractal dimension.
Waliszewski, Przemyslaw; Konarski, Jerzy
2002-03-15
The analysis of a set of experimental data obtained by an independent team of researchers confirms that neuronal differentiation or synapse formation do occur in time and space with fractal dimension. The interacting cells create first a dynamic system with its own attractor, (i.e., a fragment of time and space where the dynamic processes occur and where no further evolution of the system is possible at all owing to the action of the intrasystemic forces unless some extrasystemic forces act upon it). This attractor is then modified in the active manner by the differentiating cells until the system attains a degenerated stationary state and differentiation ends. The fractal structure of the system is also lost in the course of tumor progression. Our data indicate that the cellular system can attain the degenerated stationary state, leaving the attractor with a fractal dimension directly or undergoing diversification into many attractors and going through the areas of deterministic chaos. Since evolution of the cellular system is driven by the cooperative dynamic processes, as reflected by the changes of the mean fractal dimension between the intervals of the Gompertzian curve, it is likely that cells differentiate into neurons and create synapses with a conjugated probability and non-Gaussian distribution rather than with the classical probability and the Gaussian distribution. These findings can help to optimize features of artificial neural networks. They also define a simple in vitro biological model for biophysical and biochemical studies on natural neural networks.
Definition of fractal topography to essential understanding of scale-invariance
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.
FRACTAL PATTERN GROWTH OF METAL ATOM CLUSTERS IN ION IMPLANTED POLYMERS
Institute of Scientific and Technical Information of China (English)
ZHANG TONG-HE; WU YU-GUANG; SANG HAI-BO; ZHOU GU
2001-01-01
The fractal and multi-fractal patterns of metal atoms are observed in the surface layer and cross section of a metal ion implanted polymer using TEM and SEM for the first time. The surface structure in the metal ion implanted polyethylene terephthalane (PET) is the random fractal. Certain average quantities of the random geometric patterns contain self-similarity. Some growth origins appeared in the fractal pattern which has a dimension of 1.67. The network structure of the fractal patterns is formed in cross section, having a fractal dimension of 1.87. So it can be seen that the fractal pattern is three-dimensional space fractal. We also find the collision cascade fractal in the cross section of implanted nylon, which is similar to the collision cascade pattern in transverse view calculated by the TRIM computer program. Finally, the mechanism for the formation and growth of the fractal patterns during ion implantation is discussed.
Fractality in momentum space: A signal of criticality in nuclear collisions
Antoniou, N. G.; Davis, N.; Diakonos, F. K.
2016-01-01
We show that critical systems of finite size develop a fractal structure in momentum space with anomalous dimension given in terms of the isotherm critical exponent δ of the corresponding infinite system. The associated power laws of transverse momentum correlations, in high-energy nuclear collisions, provide us with a signature of a critical point in strongly interacting matter according to the laws of QCD.
Fractality in momentum space: a signal of criticality in nuclear collisions
Antoniou, Nikolaos G; Diakonos, Fotios K
2015-01-01
We show that critical systems of finite size develop a fractal structure in momentum space with anomalous dimension given in terms of the isotherm critical exponent delta of the corresponding infinite system. The associated power laws of transverse momentum correlations, in high-energy nuclear collisions, provide us with a signature of a critical point in strongly interacting matter according to the laws of QCD.
Neutron scattering from fractals
DEFF Research Database (Denmark)
Kjems, Jørgen; Freltoft, T.; Richter, D.
1986-01-01
The scattering formalism for fractal structures is presented. Volume fractals are exemplified by silica particle clusters formed either from colloidal suspensions or by flame hydrolysis. The determination of the fractional dimensionality through scattering experiments is reviewed, and recent small...
Mineral resource analysis by parabolic fractals
Institute of Scientific and Technical Information of China (English)
XIE Shu-yun; YANG Yong-guo; BAO Zheng-yu; KE Xian-zhong; LIU Xiao-long
2009-01-01
Elemental concentration distributions in space have been analyzed using different approaches. These analyses are of great significance for the quantitative characterization of various kinds of distribution patterns. Fractal and multi-fiactal methods have been extensively applied to this topic. Traditionally, approximately linear-fractal laws have been regarded as useful tools for characterizing the self-similarities of element concentrations. But, in nature, it is not always easy to fred perfect linear fractal laws. In this paper the parabolic fractal model is used. First a two dimensional multiplicative multi-fractal cascade model is used to study the concentration patterns. The results show the parabolic fractal (PF) properties of the concentrations and the validity of non-linear fractal analysis. By dividing the studied area into four sub-areas it was possible to show that each part follows a non-linear para-bolic fractal law and that the dispersion within each part varies. The ratio of the polynomial coefficients of the fitted parabolic curves can reflect, to some degree, the relative concentration and dispersal distribution patterns. This can provide new insight into the ore-forming potential in space. The parabolic fractal evaluations of ore-forming potential for the four subareas are in good agreement with field investigation work and geochemical mapping results based on analysis of the original data.
Projective Limits of State Spaces IV. Fractal Label Sets
Lanéry, Suzanne
2015-01-01
Instead of formulating the state space of a quantum field theory over one big Hilbert space, it has been proposed by Kijowski [Kijowski 1977] to represent quantum states as projective families of density matrices over a collection of smaller, simpler Hilbert spaces. One can thus bypass the need to select a vacuum state for the theory, and still be provided with an explicit and constructive description of the quantum state space, at least as long as the label set indexing the projective structure is countable. Because uncountable label sets are much less practical in this context, we develop in the present article a general procedure to trim an originally uncountable label set down to countable cardinality. In particular, we investigate how to perform this tightening of the label set in a way that preserves both the physical content of the algebra of observables and its symmetries. This work is notably motivated by applications to the holonomy-flux algebra underlying Loop Quantum Gravity. Building on earlier w...
Charaterisation of function spaces via mollification; fractal quantities for distributions
Directory of Open Access Journals (Sweden)
Hans Triebel
2003-01-01
Full Text Available The aim of this paper is twofold. First we characterise elements f belonging to the Besov spaces Bpqs(ℝn with s∈ℝ, 0
Indian Academy of Sciences (India)
Zitian Li
2014-09-01
A broad general variable separation solution with two arbitrary lower-dimensional functions of the (2+1)-dimensional Broer–Kaup (BK) equations was derived by means of a projective equation method and a variable separation hypothesis. Based on the derived variable separation excitation, some new special types of localized solutions such as oscillating solitons, instantonlike and cross-like fractal structures are revealed by selecting appropriate functions of the general variable separation solution.
Fractal analysis of polyferric chloride-humic acid (PFC-HA) flocs in different topological spaces.
Wang, Yili; Lu, Jia; Baiyu, Du; Shi, Baoyou; Wang, Dongsheng
2009-01-01
The fractal dimensions in different topological spaces of polyferric chloride-humic acid (PFC-HA) flocs, formed in flocculating different kinds of humic acids (HA) water at different initial pH (9.0, 7.0, 5.0) and PFC dosages, were calculated by effective density-maximum diameter, image analysis, and N2 absorption-desorption methods, respectively. The mass fractal dimensions (Df) of PFC-HA flocs were calculated by bi-logarithm relation of effective density with maximum diameter and Logan empirical equation. The Df value was more than 2.0 at initial pH of 7.0, which was 11% and 13% higher than those at pH 9.0 and 5.0, respectively, indicating the most compact flocs formed in flocculated HA water at initial pH of 7.0. The image analysis for those flocs indicates that after flocculating the HA water at initial pH greater than 7.0 with PFC flocculant, the fractal dimensions of D2 (logA vs. logdL) and D3 (logVsphere VS. logdL) of PFC-HA flocs decreased with the increase of PFC dosages, and PFC-HA flocs showed a gradually looser structure. At the optimum dosage of PFC, the D2 (logA vs. logdL) values of the flocs show 14%-43% difference with their corresponding Df, and they even had different tendency with the change of initial pH values. However, the D2 values of the flocs formed at three different initial pH in HA solution had a same tendency with the corresponding Dr. Based on fractal Frenkel-Halsey-Hill (FHH) adsorption and desorption equations, the pore surface fractal dimensions (Ds) for dried powders of PFC-HA flocs formed in HA water with initial pH 9.0 and 7.0 were all close to 2.9421, and the Ds values of flocs formed at initial pH 5.0 were less than 2.3746. It indicated that the pore surface fractal dimensions of PFC-HA flocs dried powder mainly show the irregularity from the mesopore-size distribution and marcopore-size distribution.
Four Dimensional Trace Space Measurement
Energy Technology Data Exchange (ETDEWEB)
Hernandez, M.
2005-02-10
Future high energy colliders and FELs (Free Electron Lasers) such as the proposed LCLS (Linac Coherent Light Source) at SLAC require high brightness electron beams. In general a high brightness electron beam will contain a large number of electrons that occupy a short longitudinal duration, can be focused to a small transverse area while having small transverse divergences. Therefore the beam must have a high peak current and occupy small areas in transverse phase space and so have small transverse emittances. Additionally the beam should propagate at high energy and have a low energy spread to reduce chromatic effects. The requirements of the LCLS for example are pulses which contain 10{sup 10} electrons in a temporal duration of 10 ps FWHM with projected normalized transverse emittances of 1{pi} mm mrad[1]. Currently the most promising method of producing such a beam is the RF photoinjector. The GTF (Gun Test Facility) at SLAC was constructed to produce and characterize laser and electron beams which fulfill the LCLS requirements. Emittance measurements of the electron beam at the GTF contain evidence of strong coupling between the transverse dimensions of the beam. This thesis explores the effects of this coupling on the determination of the projected emittances of the electron beam. In the presence of such a coupling the projected normalized emittance is no longer a conserved quantity. The conserved quantity is the normalized full four dimensional phase space occupied by the beam. A method to determine the presence and evaluate the strength of the coupling in emittance measurements made in the laboratory is developed. A method to calculate the four dimensional volume the beam occupies in phase space using quantities available in the laboratory environment is also developed. Results of measurements made of the electron beam at the GTF that demonstrate these concepts are presented and discussed.
Radon Transform in Finite Dimensional Hilbert Space
Revzen, M.
2012-01-01
Novel analysis of finite dimensional Hilbert space is outlined. The approach bypasses general, inherent, difficulties present in handling angular variables in finite dimensional problems: The finite dimensional, d, Hilbert space operators are underpinned with finite geometry which provide intuitive perspective to the physical operators. The analysis emphasizes a central role for projectors of mutual unbiased bases (MUB) states, extending thereby their use in finite dimensional quantum mechani...
Improved Fractal Space Filling Curves Hybrid Optimization Algorithm for Vehicle Routing Problem.
Yue, Yi-xiang; Zhang, Tong; Yue, Qun-xing
2015-01-01
Vehicle Routing Problem (VRP) is one of the key issues in optimization of modern logistics system. In this paper, a modified VRP model with hard time window is established and a Hybrid Optimization Algorithm (HOA) based on Fractal Space Filling Curves (SFC) method and Genetic Algorithm (GA) is introduced. By incorporating the proposed algorithm, SFC method can find an initial and feasible solution very fast; GA is used to improve the initial solution. Thereafter, experimental software was developed and a large number of experimental computations from Solomon's benchmark have been studied. The experimental results demonstrate the feasibility and effectiveness of the HOA.
Scale relativity and fractal space-time a new approach to unifying relativity and quantum mechanics
Nottale, Laurent
2011-01-01
This book provides a comprehensive survey of the development of the theory of scale relativity and fractal space-time. It suggests an original solution to the disunified nature of the classical-quantum transition in physical systems, enabling the basis of quantum mechanics on the principle of relativity, provided this principle is extended to scale transformations of the reference system. In the framework of such a newly generalized relativity theory (including position, orientation, motion and now scale transformations), the fundamental laws of physics may be given a general form that unifies
Improved Fractal Space Filling Curves Hybrid Optimization Algorithm for Vehicle Routing Problem
Directory of Open Access Journals (Sweden)
Yi-xiang Yue
2015-01-01
Full Text Available Vehicle Routing Problem (VRP is one of the key issues in optimization of modern logistics system. In this paper, a modified VRP model with hard time window is established and a Hybrid Optimization Algorithm (HOA based on Fractal Space Filling Curves (SFC method and Genetic Algorithm (GA is introduced. By incorporating the proposed algorithm, SFC method can find an initial and feasible solution very fast; GA is used to improve the initial solution. Thereafter, experimental software was developed and a large number of experimental computations from Solomon’s benchmark have been studied. The experimental results demonstrate the feasibility and effectiveness of the HOA.
Improved Fractal Space Filling Curves Hybrid Optimization Algorithm for Vehicle Routing Problem
Yue, Yi-xiang; Zhang, Tong; Yue, Qun-xing
2015-01-01
Vehicle Routing Problem (VRP) is one of the key issues in optimization of modern logistics system. In this paper, a modified VRP model with hard time window is established and a Hybrid Optimization Algorithm (HOA) based on Fractal Space Filling Curves (SFC) method and Genetic Algorithm (GA) is introduced. By incorporating the proposed algorithm, SFC method can find an initial and feasible solution very fast; GA is used to improve the initial solution. Thereafter, experimental software was developed and a large number of experimental computations from Solomon's benchmark have been studied. The experimental results demonstrate the feasibility and effectiveness of the HOA. PMID:26167171
Effective degrees of freedom of a random walk on a fractal.
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.
Rodriguez-Vallejo, Manuel; Monsoriu, Juan A; Ferrando, Vicente; Furlan, Walter D
2016-01-01
Purpose: To assess the peripheral refraction induced by Fractal Contact Lenses (FCLs) in myopic eyes by means of a two-dimensional Relative Peripheral Refractive Error (RPRE) map. Methods: FCLs prototypes were specially manufactured and characterized. This study involved twenty-six myopic subjects ranging from -0.50 D to -7.00 D. The two-dimensional RPRE was measured with an open-field autorefractor by means of tracking targets distributed in a square grid from -30 degrees (deg) nasal to 30 deg temporal and 15 deg superior to -15 deg inferior. Corneal topographies were taken in order to assess correlations between corneal asphericity, lens decentration and RPRE represented in vector components M, J0 and J45. Results: The mean power of the FCLs therapeutic zones was 1.32 +/- 0.28 D. Significant correlations were found between the corneal asphericity and vector components of the RPRE in the nacked eyes. FCLs were decentered a mean of 0.7 +/- 0.19 mm to the temporal cornea. M decreased asymmetrically between nas...
Three-Dimensional Surface Parameters and Multi-Fractal Spectrum of Corroded Steel.
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.
Three-Dimensional Surface Parameters and Multi-Fractal Spectrum of Corroded Steel.
Directory of Open Access Journals (Sweden)
Xu Shanhua
Full Text Available 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.
Dimensional reduction over fuzzy coset spaces
Energy Technology Data Exchange (ETDEWEB)
Aschieri, P. E-mail: aschieri@theorie.physik.uni-muenchen.de; Madore, J.; Manousselis, P.; Zoupanos, G
2004-04-01
We examine gauge theories on Minkowski space-time times fuzzy coset spaces. This means that the extra space dimensions instead of being a continuous coset space S/R are a corresponding finite matrix approximation. The gauge theory defined on this non-commutative setup is reduced to four dimensions and the rules of the corresponding dimensional reduction are established. We investigate in particular the case of the fuzzy sphere including the dimensional reduction of fermion fields. (author)
Quantum mechanics in finite dimensional Hilbert space
de la Torre, A C
2002-01-01
The quantum mechanical formalism for position and momentum of a particle in a one dimensional cyclic lattice is constructively developed. Some mathematical features characteristic of the finite dimensional Hilbert space are compared with the infinite dimensional case. The construction of an unbiased basis for state determination is discussed.
Radon Transform for Finite Dimensional Hilbert Space
Revzen, M
2012-01-01
Finite dimensional, d, Hilbert space operators are underpinned with ?nite geometry. The analysis emphasizes a central role for mutual unbiased bases (MUB) states projectors. Interrelation among the Hilbert space operators revealed via their (?nite) dual a?ne plane geometry (DAPG) underpin- ning is studied and utilized in formulating a ?nite dimensional Radon transformation. The ?nite geometry required for our study is outlines.
One Dimensional Locally Connected S-spaces
Kunen, Joan E Hart Kenneth
2007-01-01
We construct, assuming Jensen's principle diamond, a one-dimensional locally connected hereditarily separable continuum without convergent sequences. The construction is an inverse limit in omega_1 steps, and is patterned after the original Fedorchuk construction of a compact S-space. To make it one-dimensional, each space in the inverse limit is a copy of the Menger sponge.
Coset space dimensional reduction of gauge theories
Energy Technology Data Exchange (ETDEWEB)
Kapetanakis, D. (Physik Dept., Technische Univ. Muenchen, Garching (Germany)); Zoupanos, G. (CERN, Geneva (Switzerland))
1992-10-01
We review the attempts to construct unified theories defined in higher dimensions which are dimensionally reduced over coset spaces. We employ the coset space dimensional reduction scheme, which permits the detailed study of the resulting four-dimensional gauge theories. In the context of this scheme we present the difficulties and the suggested ways out in the attempts to describe the observed interactions in a realistic way. (orig.).
Using texture synthesis in fractal pattern design
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
Traditional fractal pattern design has some disadvantages such as inability to effectively reflect the characteristics of real scenery and texture. We propose a novel pattern design technique combining fractal geometry and image texture synthesis to solve these problems. We have improved Wei and Levoy (2000)'s texture synthesis algorithm by first using two-dimensional autocorrelation function to analyze the structure and distribution of textures, and then determining the size of L neighborhood.Several special fractal sets were adopted and HSL (Hue, Saturation, and Light) color space was chosen. The fractal structure was used to manipulate the texture synthesis in HSL color space where the pattern's color can be adjusted conveniently. Experiments showed that patterns with different styles and different color characteristics can be more efficiently generated using the new technique.
Physical model of dimensional regularization
Energy Technology Data Exchange (ETDEWEB)
Schonfeld, Jonathan F.
2016-12-15
We explicitly construct fractals of dimension 4-ε on which dimensional regularization approximates scalar-field-only quantum-field theory amplitudes. The construction does not require fractals to be Lorentz-invariant in any sense, and we argue that there probably is no Lorentz-invariant fractal of dimension greater than 2. We derive dimensional regularization's power-law screening first for fractals obtained by removing voids from 3-dimensional Euclidean space. The derivation applies techniques from elementary dielectric theory. Surprisingly, fractal geometry by itself does not guarantee the appropriate power-law behavior; boundary conditions at fractal voids also play an important role. We then extend the derivation to 4-dimensional Minkowski space. We comment on generalization to non-scalar fields, and speculate about implications for quantum gravity. (orig.)
Geometrical Underpinning of Finite Dimensional Hilbert space
Revzen, M
2011-01-01
Finite geometry is employed to underpin operators in finite, d, dimensional Hilbert space. The central role of Hilbert space operators that form mutual unbiased bases (MUB) states projectors is exhibited. Interrelation among them revealed through their (finite) dual affine plane geometry (DAPG) underpinning is studied. Transcription to (finite) affine plane geometry (APG) is given and utilized for their interpretation.
Geometrical Underpinning of Finite Dimensional Hilbert space
Revzen, M.
2011-01-01
Finite geometry is employed to underpin operators in finite, d, dimensional Hilbert space. The central role of mutual unbiased bases (MUB) states projectors is exhibited. Interrelation among operators in Hilbert space, revealed through their (finite) dual affine plane geometry (DAPG) underpinning is studied. Transcription to (finite) affine plane geometry (APG) is given and utilized for their interpretation.
Riemann zeros, prime numbers, and fractal potentials.
van Zyl, Brandon P; Hutchinson, David A W
2003-06-01
Using two distinct inversion techniques, the local one-dimensional potentials for the Riemann zeros and prime number sequence are reconstructed. We establish that both inversion techniques, when applied to the same set of levels, lead to the same fractal potential. This provides numerical evidence that the potential obtained by inversion of a set of energy levels is unique in one dimension. We also investigate the fractal properties of the reconstructed potentials and estimate the fractal dimensions to be D=1.5 for the Riemann zeros and D=1.8 for the prime numbers. This result is somewhat surprising since the nearest-neighbor spacings of the Riemann zeros are known to be chaotically distributed, whereas the primes obey almost Poissonlike statistics. Our findings show that the fractal dimension is dependent on both level statistics and spectral rigidity, Delta(3), of the energy levels.
Fractal dynamics in chaotic quantum transport.
Kotimäki, V; Räsänen, E; Hennig, H; Heller, E J
2013-08-01
Despite several experiments on chaotic quantum transport in two-dimensional systems such as semiconductor quantum dots, corresponding quantum simulations within a real-space model have been out of reach so far. Here we carry out quantum transport calculations in real space and real time for a two-dimensional stadium cavity that shows chaotic dynamics. By applying a large set of magnetic fields we obtain a complete picture of magnetoconductance that indicates fractal scaling. In the calculations of the fractality we use detrended fluctuation analysis-a widely used method in time-series analysis-and show its usefulness in the interpretation of the conductance curves. Comparison with a standard method to extract the fractal dimension leads to consistent results that in turn qualitatively agree with the previous experimental data.
Relativistic phase space dimensional recurrences
Delbourgo, Robert
2003-01-01
We derive recurrence relations between phase space expressions in different dimensions by confining some of the coordinates to tori or spheres of radius $R$ and taking the limit as $R \\to \\infty$. These relations take the form of mass integrals, associated with extraneous momenta (relative to the lower dimension), and produce the result in the higher dimension.
Photonic Band Gaps in Two-Dimensional Crystals with Fractal Structure
Institute of Scientific and Technical Information of China (English)
刘征; 徐建军; 林志方
2003-01-01
We simulate the changes of the photonic band structure of the crystal in two dimensions with a quasi-fractal structure when it is fined to a fractal. The result shows that when the dielectric distribution is fined, the photonic band structure will be compressed on the whole and the ground photonic band gap (PBG) closed while the next PBGs shrunk, in conjunction with their position declining in the frequency spectrum. Furthermore, the PBGs in the high zone are much more sensitive than those in low zones.
A Note on 3 + 1 Dimensionality
Sidharth, B G
2000-01-01
Recent work by Castro, Granik and El Naschie has given a rationale for the three dimensionality of our physical space within the framework of cantorian fractal space time using similar ideas of quantized fractal space time and noncommutativity. We also deduce the same result. Interestingly this is also seen to provide a rationale for an unproven conjecture of Poincare.
Institute of Scientific and Technical Information of China (English)
冯春华; 刘力; 刘守忠; 宁红; 孙海坚; 郭爱克
1995-01-01
The optical recording of three-dimensional(3-D)reconstruction of CA1 pyramidal cells wasderived from the studies on the CA1 region of the hippocampus in adult male Wistar rats.The recordingwas produced by the Confocal Laser Scan Microscope(LSM-10).The attemption was to outline themorphological neural network of CA1 pyramidal cells organization,following the trail of axo-dendritic connec-tions in 3-D spatial distributions among neurons.The fractal structure of neurons with their dendritic andaxonal trees using fractal algorithm was noticed,and 2—18 simulated cells were obtained using PC-486 comput-er.The simulational cells are similar in morphology to the natural CA1 hippocampal pyramidal cells.There-fore,the exploitation of an advanced neurohistological research technique combining optical recording of theLSM-10 and computer simulation of fractal structure can provide the quantitative fractal structural basis forchaosic dynamics of brain.
Jarraya, Mohamed; Guermazi, Ali; Niu, Jingbo; Duryea, Jeffrey; Lynch, John A; Roemer, Frank W
2015-11-01
The aim of this study has been to test reproducibility of fractal signature analysis (FSA) in a young, active patient population taking into account several parameters including intra- and inter-reader placement of regions of interest (ROIs) as well as various aspects of projection geometry. In total, 685 patients were included (135 athletes and 550 non-athletes, 18-36 years old). Regions of interest (ROI) were situated beneath the medial tibial plateau. The reproducibility of texture parameters was evaluated using intraclass correlation coefficients (ICC). Multi-dimensional assessment included: (1) anterior-posterior (A.P.) vs. posterior-anterior (P.A.) (Lyon-Schuss technique) views on 102 knees; (2) unilateral (single knee) vs. bilateral (both knees) acquisition on 27 knees (acquisition technique otherwise identical; same A.P. or P.A. view); (3) repetition of the same image acquisition on 46 knees (same A.P. or P.A. view, and same unitlateral or bilateral acquisition); and (4) intra- and inter-reader reliability with repeated placement of the ROIs in the subchondral bone area on 99 randomly chosen knees. ICC values on the reproducibility of texture parameters for A.P. vs. P.A. image acquisitions for horizontal and vertical dimensions combined were 0.72 (95% confidence interval (CI) 0.70-0.74) ranging from 0.47 to 0.81 for the different dimensions. For unilateral vs. bilateral image acquisitions, the ICCs were 0.79 (95% CI 0.76-0.82) ranging from 0.55 to 0.88. For the repetition of the identical view, the ICCs were 0.82 (95% CI 0.80-0.84) ranging from 0.67 to 0.85. Intra-reader reliability was 0.93 (95% CI 0.92-0.94) and inter-observer reliability was 0.96 (95% CI 0.88-0.99). A decrease in reliability was observed with increasing voxel sizes. Our study confirms excellent intra- and inter-reader reliability for FSA, however, results seem to be affected by acquisition technique, which has not been previously recognized.
Four dimensional trace space measurement
Hernández, M E; Winick, Herman; Smith, T
2003-01-01
Future high energy colliders and FELs (Free Electron Lasers) such as the proposed LCLS (Linac Coherent Light Source) at SLAC require high brightness electron beams. In general a high brightness electron beam will contain a large number of electrons that occupy a short longitudinal duration, can be focused to a small transverse area while having small transverse divergences. Therefore the beam must have a high peak current and occupy small areas in transverse phase space and so have small transverse emittances. Additionally the beam should propagate at high energy and have a low energy spread to reduce chromatic effects. The requirements of the LCLS for example are pulses which contain 1010 electrons in a temporal duration of 10 ps FWHM with projected normalized transverse emittances of 1π mm mrad. Currently the most promising method of producing such a beam is the RF photoinjector. The GTF (Gun Test Facility) at SLAC was constructed to produce and characterize laser and electron beams which fulfill the...
Khokhlov, D L
1999-01-01
The model of the universe is considered in which background of the universe is not defined by the matter but is a priori specified as a homogenous and isotropic flat space. The scale factor of the universe follows the linear law. The scale of mass changes proportional to the scale factor. This leads to that the universe has the fractal structure with a power index of 2.
Fractal Solutions of the Nizhnik-Novikov-Veselov Equation
Institute of Scientific and Technical Information of China (English)
楼森岳; 唐晓艳; 陈春丽
2002-01-01
Considering that some types of fractal solutions may appear in many (2+ l )-dimensional soliton equations because some arbitrary functions can be included in the exact solutions, we use some special types of lower dimensional fractal functions to construct higher dimensional fractal solutions of the Nizhnik Novikov-Veselov equation. The static eagle-shape fractal solutions, fractal dromion solutions and the fractal lump solutions are given in detail.
Exotic topological order from quantum fractal code
Yoshida, Beni
2014-03-01
We present a large class of three-dimensional spin models that possess topological order with stability against local perturbations, but are beyond description of topological quantum field theory. Conventional topological spin liquids, on a formal level, may be viewed as condensation of string-like extended objects with discrete gauge symmetries, being at fixed points with continuous scale symmetries. In contrast, ground states of fractal spin liquids are condensation of highly-fluctuating fractal objects with certain algebraic symmetries, corresponding to limit cycles under real-space renormalization group transformations which naturally arise from discrete scale symmetries of underlying fractal geometries. A particular class of three-dimensional models proposed in this paper may potentially saturate quantum information storage capacity for local spin systems.
Exotic topological order in fractal spin liquids
Yoshida, Beni
2013-09-01
We present a large class of three-dimensional spin models that possess topological order with stability against local perturbations, but are beyond description of topological quantum field theory. Conventional topological spin liquids, on a formal level, may be viewed as condensation of stringlike extended objects with discrete gauge symmetries, being at fixed points with continuous scale symmetries. In contrast, ground states of fractal spin liquids are condensation of highly fluctuating fractal objects with certain algebraic symmetries, corresponding to limit cycles under real-space renormalization group transformations which naturally arise from discrete scale symmetries of underlying fractal geometries. A particular class of three-dimensional models proposed in this paper may potentially saturate quantum information storage capacity for local spin systems.
Random-fractal Ansatz for the configurations of two-dimensional critical systems
Lee, Ching Hua; Ozaki, Dai; Matsueda, Hiroaki
2016-12-01
Critical systems have always intrigued physicists and precipitated the development of new techniques. Recently, there has been renewed interest in the information contained in the configurations of classical critical systems, whose computation do not require full knowledge of the wave function. Inspired by holographic duality, we investigated the entanglement properties of the classical configurations (snapshots) of the Potts model by introducing an Ansatz ensemble of random fractal images. By virtue of the central limit theorem, our Ansatz accurately reproduces the entanglement spectra of actual Potts snapshots without any fine tuning of parameters or artificial restrictions on ensemble choice. It provides a microscopic interpretation of the results of previous studies, which established a relation between the scaling behavior of snapshot entropy and the critical exponent. More importantly, it elucidates the role of ensemble disorder in restoring conformal invariance, an aspect previously ignored. Away from criticality, the breakdown of scale invariance leads to a renormalization of the parameter Σ in the random fractal Ansatz, whose variation can be used as an alternative determination of the critical exponent. We conclude by providing a recipe for the explicit construction of fractal unit cells consistent with a given scaling exponent.
El Naschie's {epsilon} {sup ({infinity})} space-time and new results in scale relativity theories
Energy Technology Data Exchange (ETDEWEB)
Gottlieb, I. [' Al.I.Cuza' University, Faculty of Physics, Department of Theoretical Physics, Blvd. Carol No. 1, Iasi 700506 (Romania)]. E-mail: gottlieb@uaic.ro; Agop, M. [' Gh. Asachi' Technical University, Department of Physics, Blvd. Mangeron, No. 64, Iasi 700050 (Romania); Ciobanu, Gabriela [' Al.I.Cuza' University, Faculty of Physics, Department of Theoretical Physics, Blvd. Carol No. 1, Iasi 700506 (Romania); Stroe, Aurelia [' Al.I.Cuza' University, Faculty of Physics, Department of Theoretical Physics, Blvd. Carol No. 1, Iasi 700506 (Romania)
2006-10-15
New results in fractal space-time theory are established: the fractal operator for the fractal dimension D = 2 implies a generalized Schroedinger equation in the Nottale's scale relativity theory, while the fractal operator for the fractal dimension D = 3 implies the Korteweg-de Vries equation in the one-dimensional case. The connection with El Naschie's {epsilon} {sup ({infinity})} theory result by means of the Cooper-type pairs or through wave-particle duality.
Why Observable Space Is Solely Three Dimensional
Rabinowitz, Mario
2015-01-01
Quantum (and classical) binding energy considerations in n-dimensional space indicate that atoms (and planets) can only exist in three-dimensional space. This is why observable space is solely 3-dimensional. Both a novel Virial theorem analysis, and detailed classical and quantum energy calculations for 3-space circular and elliptical orbits indicate that they have no orbital binding energy in greater than 3-space. The same energy equation also excludes the possibility of atom-like bodies in strictly 1 and 2-dimensions. A prediction is made that in the search for deviations from r^-2 of the gravitational force at sub-millimeter distances such a deviation must occur at < ~ 10^-10 m (or < ~10^-12 m considering muoniom), since atoms would disintegrate if the curled up dimensions of string theory were larger than this. Callender asserts that the often-repeated claim in previous work that stable orbits are possible in only three dimensions is not even remotely established. The binding energy analysis herein ...
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)
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)
Geometry of Time and Dimensionality of Space
Saniga, M
2003-01-01
One of the most distinguished features of our algebraic geometrical, pencil concept of space-time is the fact that spatial dimensions and time stand, as far as their intrinsic structure is concerned, on completely different footings: the former being represented by pencils of lines, the latter by a pencil of conics. As a consequence, we argue that even at the classical (macroscopic) level there exists a much more intricate and profound coupling between space and time than that dictated by (general) relativity theory. It is surmised that this coupling can be furnished by so-called Cremona (or birational) transformations between two projective spaces of three dimensions, being fully embodied in the structure of configurations of their fundamental elements. We review properties of some of the simplest Cremona transformations and show that the corresponding "fundamental" space-times exhibit an intimate connection between the extrinsic geometry of time dimension and the dimensionality of space. Moreover, these Cre...
Baryshev, Yuri
2002-01-01
This is the first book to present the fascinating new results on the largest fractal structures in the universe. It guides the reader, in a simple way, to the frontiers of astronomy, explaining how fractals appear in cosmic physics, from our solar system to the megafractals in deep space. It also offers a personal view of the history of the idea of self-similarity and of cosmological principles, from Plato's ideal architecture of the heavens to Mandelbrot's fractals in the modern physical cosmos. In addition, this invaluable book presents the great fractal debate in astronomy (after Luciano Pi
Escape dynamics and fractal basins boundaries in the three-dimensional Earth-Moon system
Zotos, Euaggelos E
2016-01-01
The orbital dynamics of a spacecraft, or a comet, or an asteroid in the Earth-Moon system in a scattering region around the Moon using the three dimensional version of the circular restricted three-body problem is numerically investigated. The test particle can move in bounded orbits around the Moon or escape through the openings around the Lagrange points $L_1$ and $L_2$ or even collide with the surface of the Moon. We explore in detail the first four of the five possible Hill's regions configurations depending on the value of the Jacobi constant which is of course related with the total orbital energy. We conduct a thorough numerical analysis on the phase space mixing by classifying initial conditions of orbits in several two-dimensional types of planes and distinguishing between four types of motion: (i) ordered bounded, (ii) trapped chaotic, (iii) escaping and (iv) collisional. In particular, we locate the different basins and we relate them with the corresponding spatial distributions of the escape and c...
Energy Technology Data Exchange (ETDEWEB)
Conte, Elio [Department of Pharmacology and Human Physiology and Tires, Center for Innovative Technologies for Signal Detection and Processing, University of Bari (Italy); School of Advanced International Studies on Theoretical and Nonlinear Methodologies-Bari (Italy)], E-mail: elio.conte@fastwebnet.it; Khrennikov, Andrei [International Center for Mathematical Modelling in Physics and Cognitive Sciences, M.S.I., University of Vaexjoe, S-35195 (Sweden); Federici, Antonio [Department of Pharmacology and Human Physiology and Tires, Center for Innovative Technologies for Signal Detection and Processing, University of Bari (Italy); Zbilut, Joseph P. [Department of Molecular Biophysics and Physiology, Rush University Medical Center, 1653W Congress, Chicago, IL 60612 (United States)
2009-09-15
We develop a new method for analysis of fundamental brain waves as recorded by the EEG. To this purpose we introduce a Fractal Variance Function that is based on the calculation of the variogram. The method is completed by using Random Matrix Theory. Some examples are given. We also discuss the link of such formulation with H. Weiss and V. Weiss golden ratio found in the brain, and with El Naschie fractal Cantorian space-time theory.
Fractal Dimension in Epileptic EEG Signal Analysis
Uthayakumar, R.
Fractal Analysis is the well developed theory in the data analysis of non-linear time series. Especially Fractal Dimension is a powerful mathematical tool for modeling many physical and biological time signals with high complexity and irregularity. Fractal dimension is a suitable tool for analyzing the nonlinear behaviour and state of the many chaotic systems. Particularly in analysis of chaotic time series such as electroencephalograms (EEG), this feature has been used to identify and distinguish specific states of physiological function.Epilepsy is the main fatal neurological disorder in our brain, which is analyzed by the biomedical signal called Electroencephalogram (EEG). The detection of Epileptic seizures in the EEG Signals is an important tool in the diagnosis of epilepsy. So we made an attempt to analyze the EEG in depth for knowing the mystery of human consciousness. EEG has more fluctuations recorded from the human brain due to the spontaneous electrical activity. Hence EEG Signals are represented as Fractal Time Series.The algorithms of fractal dimension methods have weak ability to the estimation of complexity in the irregular graphs. Divider method is widely used to obtain the fractal dimension of curves embedded into a 2-dimensional space. The major problem is choosing initial and final step length of dividers. We propose a new algorithm based on the size measure relationship (SMR) method, quantifying the dimensional behaviour of irregular rectifiable graphs with minimum time complexity. The evidence for the suitability (equality with the nature of dimension) of the algorithm is illustrated graphically.We would like to demonstrate the criterion for the selection of dividers (minimum and maximum value) in the calculation of fractal dimension of the irregular curves with minimum time complexity. For that we design a new method of computing fractal dimension (FD) of biomedical waveforms. Compared to Higuchi's algorithm, advantages of this method include
Critical Behavior of Gaussian Model on X Fractal Lattices in External Magnetic Fields
Institute of Scientific and Technical Information of China (English)
LI Ying; KONG Xiang-Mu; HUANG Jia-Yin
2003-01-01
Using the renormalization group method, the critical behavior of Gaussian model is studied in external magnetic fields on X fractal lattices embedded in two-dimensional and d-dimensional (d ＞ 2) Euclidean spaces, respectively. Critical points and exponents are calculated. It is found that there is long-range order at finite temperature for this model, and that the critical points do not change with the space dimensionality d (or the fractal dimensionality dr). It is also found that the critical exponents are very different from results of Ising model on the same lattices, and that the exponents on X lattices are different from the exact results on translationally symmetric lattices.
Execution spaces for simple higher dimensional automata
DEFF Research Database (Denmark)
Raussen, Martin
Higher Dimensional Automata (HDA) are highly expressive models for concurrency in Computer Science, cf van Glabbeek [26]. For a topologist, they are attractive since they can be modeled as cubical complexes - with an inbuilt restriction for directions´of allowable (d-)paths. In Raussen [25], we...... applicable in greater generality. Furthermore, we take a close look at semaphore models with semaphores all of arity one. It turns out that execution spaces for these are always homotopy discrete with components representing sets of “compatible” permutations. Finally, we describe a model for the complement...
Energy Technology Data Exchange (ETDEWEB)
Goh, Vicky; Sanghera, Bal [Mount Vernon Hospital, Paul Strickland Scanner Centre, Northwood, Middlesex (United Kingdom); Wellsted, David M.; Sundin, Josefin [University of Hertfordshire, Research and Development Support Unit, Hatfield (United Kingdom); Halligan, Steve [University College Hospital, Department of Academic Radiology, London (United Kingdom)
2009-06-15
The aim was to evaluate the feasibility of fractal analysis for assessing the spatial pattern of colorectal tumour perfusion at dynamic contrast-enhanced CT (perfusion CT). Twenty patients with colorectal adenocarcinoma underwent a 65-s perfusion CT study from which a perfusion parametric map was generated using validated commercial software. The tumour was identified by an experienced radiologist, segmented via thresholding and fractal analysis applied using in-house software: fractal dimension, abundance and lacunarity were assessed for the entire outlined tumour and for selected representative areas within the tumour of low and high perfusion. Comparison was made with ten patients with normal colons, processed in a similar manner, using two-way mixed analysis of variance with statistical significance at the 5% level. Fractal values were higher in cancer than normal colon (p {<=} 0.001): mean (SD) 1.71 (0.07) versus 1.61 (0.07) for fractal dimension and 7.82 (0.62) and 6.89 (0.47) for fractal abundance. Fractal values were lower in 'high' than 'low' perfusion areas. Lacunarity curves were shifted to the right for cancer compared with normal colon. In conclusion, colorectal cancer mapped by perfusion CT demonstrates fractal properties. Fractal analysis is feasible, potentially providing a quantitative measure of the spatial pattern of tumour perfusion. (orig.)
Transition of Urban Space in Two Systems : the Fractal Geometry of Hungarian Provincial Cities
Directory of Open Access Journals (Sweden)
E. Nagy
2000-11-01
Full Text Available En partant d'une analyse des transformations intervenues dans l'urbanisme et les fonctions des villes hongroises depuis le début des années 1990, une étude de morphologie urbaine est conduite sur les cas de Debrecen et de Szeged et illustrée par des mesures fractales pour les cinquante dernières années.
Directory of Open Access Journals (Sweden)
Cahill R. T.
2012-07-01
Full Text Available Experiments have revealed that the Fresnel drag effect is not present in RF coaxial cables, contrary to a previous report. This enables a very sensitive, robust and compact detector, that is 1st order in v / c and using one clock, to detect the dynamical space passing the earth, revealing the sidereal rotation of the earth, together with significant wave / turbulence e ff ects. These are “gravitational waves”, and previously detected by Cahill 2006, using an Optical-Fibre – RF Coaxial Cable Detector, and Cahill 2009, using a preliminary version of the Dual RF Coaxial Cable Detector. The gravitational waves have a 1 / f spectrum, implying a fractal structure to the textured dynamical 3- space.
Chaotic Maps Dynamics, Fractals, and Rapid Fluctuations
Chen, Goong
2011-01-01
This book consists of lecture notes for a semester-long introductory graduate course on dynamical systems and chaos taught by the authors at Texas A&M University and Zhongshan University, China. There are ten chapters in the main body of the book, covering an elementary theory of chaotic maps in finite-dimensional spaces. The topics include one-dimensional dynamical systems (interval maps), bifurcations, general topological, symbolic dynamical systems, fractals and a class of infinite-dimensional dynamical systems which are induced by interval maps, plus rapid fluctuations of chaotic maps as a
McAteer, R. T. J.
2013-06-01
When Mandelbrot, the father of modern fractal geometry, made this seemingly obvious statement he was trying to show that we should move out of our comfortable Euclidean space and adopt a fractal approach to geometry. The concepts and mathematical tools of fractal geometry provides insight into natural physical systems that Euclidean tools cannot do. The benet from applying fractal geometry to studies of Self-Organized Criticality (SOC) are even greater. SOC and fractal geometry share concepts of dynamic n-body interactions, apparent non-predictability, self-similarity, and an approach to global statistics in space and time that make these two areas into naturally paired research techniques. Further, the iterative generation techniques used in both SOC models and in fractals mean they share common features and common problems. This chapter explores the strong historical connections between fractal geometry and SOC from both a mathematical and conceptual understanding, explores modern day interactions between these two topics, and discusses how this is likely to evolve into an even stronger link in the near future.
Directory of Open Access Journals (Sweden)
M. A. Navascués
2013-01-01
Full Text Available This paper tackles the construction of fractal maps on the unit sphere. The functions defined are a generalization of the classical spherical harmonics. The methodology used involves an iterated function system and a linear and bounded operator of functions on the sphere. For a suitable choice of the coefficients of the system, one obtains classical maps on the sphere. The different values of the system parameters provide Bessel sequences, frames, and Riesz fractal bases for the Lebesgue space of the square integrable functions on the sphere. The Laplace series expansion is generalized to a sum in terms of the new fractal mappings.
Directory of Open Access Journals (Sweden)
Cahill R. T.
2006-01-01
Full Text Available The new dynamical “quantum foam” theory of 3-space is described at the classical level by a velocity field. This has been repeatedly detected and for which the dynamical equations are now established. These equations predict 3-space “gravitational wave” effects, and these have been observed, and the 1991 DeWitte data is analysed to reveal the fractal structure of these “gravitational waves”. This velocity field describes the differential motion of 3-space, and the various equations of physics must be generalised to incorporate this 3-space dynamics. Here a new generalised Schrödinger equation is given and analysed. It is shown that from this equation the equivalence principle may be derived as a quantum effect, and that as well this generalised Schrödinger equation determines the effects of vorticity of the 3-space flow, or “frame-dragging”, on matter, and which is being studied by the Gravity Probe B (GP-B satellite gyroscope experiment.
Band structure characteristics of T-square fractal phononic crystals
Institute of Scientific and Technical Information of China (English)
Liu Xiao-Jian; Fan You-Hua
2013-01-01
The T-square fractal two-dimensional phononic crystal model is presented in this article.A comprehensive study is performed for the Bragg scattering and locally resonant fractal phononic crystal.We find that the band structures of the fractal and non-fractal phononic crystals at the same filling ratio are quite different through using the finite element method.The fractal design has an important impact on the band structures of the two-dimensional phononic crystals.
Zhao, Leihong; Yang, Lining; Lin, Hongjun; Zhang, Meijia; Yu, Haiying; Liao, Bao-Qiang; Wang, Fangyuan; Zhou, Xiaoling; Li, Renjie
2016-12-01
While the adsorptive fouling in membrane bioreactors (MBRs) is highly dependent of the surface morphology, little progress has been made on modeling biocake layer surface morphology. In this study, a novel method, which combined static light scattering method for fractal dimension (Df) measurement with fractal method represented by the modified two-variable Weierstrass-Mandelbrot function, was proposed to model biocake layer surface in a MBR. Characterization by atomic force microscopy showed that the biocake surface was stochastic, disorder, self-similarity, and with non-integer dimension, illustrating obvious fractal features. Fractal dimension (Df) of sludge suspension experienced a significant change with operation of the MBR. The constructed biocake layer surface by the proposed method was quite close to the real surface, showing the feasibility of the proposed method. It was found that Df was the critical factor affecting surface morphology, while other factors exerted moderate or minor effects on the roughness of biocake layer.
Charged fluid distribution in higher dimensional spheroidal space-time
Indian Academy of Sciences (India)
G P Singh; S Kotambkar
2005-07-01
A general solution of Einstein field equations corresponding to a charged fluid distribution on the background of higher dimensional spheroidal space-time is obtained. The solution generates several known solutions for superdense star having spheroidal space-time geometry.
-Boundedness and -Compactness in Finite Dimensional Probabilistic Normed Spaces
Indian Academy of Sciences (India)
Reza Saadati; Massoud Amini
2005-11-01
In this paper, we prove that in a finite dimensional probabilistic normed space, every two probabilistic norms are equivalent and we study the notion of -compactness and -boundedness in probabilistic normed spaces.
Tremblay, A.-M. S.; Breton, P.
1984-03-01
The application of exact and approximate position-space renormalization group techniques to the calculation of densities of states for problems with Gaussian generating functions (such as free tight-binding electrons, harmonic vibrations, spin waves, or random walks on Euclidian or 'fractal' lattices) is briefly reviewed. It is also shown that for one-dimensional Gaussian theories with disorder, the approximate recursion relations proposed by Goncalves da Silva and Koiller (GK) are exact for problems formulated on Berker-Ostlund lattices. A generalization of the GK scheme which allows one to calculate the optical zone-center density of states is formulated and then applied to the study of oneand two-mode behavior in mixed diatomic crystals.
Fractal structures in centrifugal flywheel governor system
Rao, Xiao-Bo; Chu, Yan-Dong; Lu-Xu; Chang, Ying-Xiang; Zhang, Jian-Gang
2017-09-01
The global structure of nonlinear response of mechanical centrifugal governor, forming in two-dimensional parameter space, is studied in this paper. By using three kinds of phases, we describe how responses of periodicity, quasi-periodicity and chaos organize some self-similarity structures with parameters varying. For several parameter combinations, the regular vibration shows fractal characteristic, that is, the comb-shaped self-similarity structure is generated by alternating periodic response with intermittent chaos, and Arnold's tongues embedded in quasi-periodic response are organized according to Stern-Brocot tree. In particular, a new type of mixed-mode oscillations (MMOs) is found in the periodic response. These unique structures reveal the natural connection of various responses between part and part, part and the whole in parameter space based on self-similarity of fractal. Meanwhile, the remarkable and unexpected results are to contribute a valid dynamic reference for practical applications with respect to mechanical centrifugal governor.
Configuration entropy of fractal landscapes
National Research Council Canada - National Science Library
Rodríguez‐Iturbe, Ignacio; D'Odorico, Paolo; Rinaldo, Andrea
1998-01-01
.... The spatial arrangement of two‐dimensional images is found to be an effective way to characterize fractal landscapes and the configurational entropy of these arrangements imposes demanding conditions for models attempting to represent these fields.
Analysis of fractals with combined partition
Dedovich, T. G.; Tokarev, M. V.
2016-03-01
The space—time properties in the general theory of relativity, as well as the discreteness and non-Archimedean property of space in the quantum theory of gravitation, are discussed. It is emphasized that the properties of bodies in non-Archimedean spaces coincide with the properties of the field of P-adic numbers and fractals. It is suggested that parton showers, used for describing interactions between particles and nuclei at high energies, have a fractal structure. A mechanism of fractal formation with combined partition is considered. The modified SePaC method is offered for the analysis of such fractals. The BC, PaC, and SePaC methods for determining a fractal dimension and other fractal characteristics (numbers of levels and values of a base of forming a fractal) are considered. It is found that the SePaC method has advantages for the analysis of fractals with combined partition.
Visible parts of fractal percolation
Arhosalo, I; Järvenpää, M; Rams, M; Shmerkin, P
2009-01-01
We study dimensional properties of visible parts of fractal percolation in the plane. Provided that the dimension of the fractal percolation is at least 1, we show that, conditioned on non-extinction, almost surely all visible parts from lines are 1-dimensional. Furthermore, almost all of them have positive and finite Hausdorff measure. We also verify analogous results for visible parts from points. These results are motivated by an open problem on the dimensions of visible parts.
On the consistency of coset space dimensional reduction
Energy Technology Data Exchange (ETDEWEB)
Chatzistavrakidis, A. [Institute of Nuclear Physics, NCSR DEMOKRITOS, GR-15310 Athens (Greece); Physics Department, National Technical University of Athens, GR-15780 Zografou Campus, Athens (Greece)], E-mail: cthan@mail.ntua.gr; Manousselis, P. [Physics Department, National Technical University of Athens, GR-15780 Zografou Campus, Athens (Greece); Department of Engineering Sciences, University of Patras, GR-26110 Patras (Greece)], E-mail: pman@central.ntua.gr; Prezas, N. [CERN PH-TH, 1211 Geneva (Switzerland)], E-mail: nikolaos.prezas@cern.ch; Zoupanos, G. [Physics Department, National Technical University of Athens, GR-15780 Zografou Campus, Athens (Greece)], E-mail: george.zoupanos@cern.ch
2007-11-15
In this Letter we consider higher-dimensional Yang-Mills theories and examine their consistent coset space dimensional reduction. Utilizing a suitable ansatz and imposing a simple set of constraints we determine the four-dimensional gauge theory obtained from the reduction of both the higher-dimensional Lagrangian and the corresponding equations of motion. The two reductions yield equivalent results and hence they constitute an example of a consistent truncation.
Dewdney, A. K.
1991-01-01
Explores the subject of fractal geometry focusing on the occurrence of fractal-like shapes in the natural world. Topics include iterated functions, chaos theory, the Lorenz attractor, logistic maps, the Mandelbrot set, and mini-Mandelbrot sets. Provides appropriate computer algorithms, as well as further sources of information. (JJK)
Osler, Thomas J.
1999-01-01
Because fractal images are by nature very complex, it can be inspiring and instructive to create the code in the classroom and watch the fractal image evolve as the user slowly changes some important parameter or zooms in and out of the image. Uses programming language that permits the user to store and retrieve a graphics image as a disk file.…
Directory of Open Access Journals (Sweden)
Tatjana eStadnitski
2012-05-01
Full Text Available When investigating fractal phenomena, the following questions are fundamental for the applied researcher: (1 What are essential statistical properties of 1/f noise? (2 Which estimators are available for measuring fractality? (3 Which measurement instruments are appropriate and how are they applied? The purpose of this article is to give clear and comprehensible answers to these questions. First, theoretical characteristics of a fractal pattern (self-similarity, long memory, power law and the related fractal parameters (the Hurst coefficient, the scaling exponent, the fractional differencing parameter d of the ARFIMA methodology, the power exponent of the spectral analysis are discussed. Then, estimators of fractal parameters from different software packages commonly used by applied researchers (R, SAS, SPSS are introduced and evaluated. Advantages, disadvantages, and constrains of the popular estimators are illustrated by elaborate examples. Finally, crucial steps of fractal analysis (plotting time series data, autocorrelation and spectral functions; performing stationarity tests; choosing an adequate estimator; estimating fractal parameters; distinguishing fractal processes from short memory patterns are demonstrated with empirical time series.
Stadnitski, Tatjana
2012-01-01
WHEN INVESTIGATING FRACTAL PHENOMENA, THE FOLLOWING QUESTIONS ARE FUNDAMENTAL FOR THE APPLIED RESEARCHER: (1) What are essential statistical properties of 1/f noise? (2) Which estimators are available for measuring fractality? (3) Which measurement instruments are appropriate and how are they applied? The purpose of this article is to give clear and comprehensible answers to these questions. First, theoretical characteristics of a fractal pattern (self-similarity, long memory, power law) and the related fractal parameters (the Hurst coefficient, the scaling exponent α, the fractional differencing parameter d of the autoregressive fractionally integrated moving average methodology, the power exponent β of the spectral analysis) are discussed. Then, estimators of fractal parameters from different software packages commonly used by applied researchers (R, SAS, SPSS) are introduced and evaluated. Advantages, disadvantages, and constrains of the popular estimators ([Formula: see text] power spectral density, detrended fluctuation analysis, signal summation conversion) are illustrated by elaborate examples. Finally, crucial steps of fractal analysis (plotting time series data, autocorrelation, and spectral functions; performing stationarity tests; choosing an adequate estimator; estimating fractal parameters; distinguishing fractal processes from short-memory patterns) are demonstrated with empirical time series.
On the fractal properties microaccelerations
Sedelnikov, A V
2012-01-01
In this paper the fractal property of the internal environment of space laboratory microaccelerations that occur. Changing the size of the space lab leads to the fact that the dependence of microaccelerations from time to time has the property similar to the self-affinity of fractal functions. With the help of microaccelerations, based on the model of the real part of the fractal Weierstrass-Mandelbrot function is proposed to form the inertial-mass characteristics of laboratory space with a given level of microaccelerations.
Measures and dimensions of fractal sets in local fields
Institute of Scientific and Technical Information of China (English)
QIU Hua; SU Weiyi
2006-01-01
The study of fractal analysis over the local fields as underline spaces is very important since it can motivate new approaches and new ideas, and discover new techniques in the study of fractals. To study fractal sets in a local field K, in this paper, we define several kinds of fractal measures and dimensions of subsets in K. Some typical fractal sets in K are constructed. We also give out the Hausdorff dimensions and measures, Box-counting dimensions and Packing dimensions, and stress that there exist differences between fractal analysis on local fields and Euclidean spaces. Consequently, the theoretical foundation of fractal analysis on local fields is established.
Crossover and thermodynamic representation in the extended η model for fractal growth
Nagatani, Takashi; Stanley, H. Eugene
1990-10-01
The η model for the dielectric breakdown is extended to the case where double power laws apply. It is shown that a crossover phenomenon between the diffusion-limited aggregation (DLA) fractal and the η fractal occurs in the extended η model. Through the use of the dimensional analysis, a dimensionless parameter is found to govern the crossover. It is shown that when η1 the inverse crossover from the η fractal to the DLA fractal appears. It is also shown that the crossover radius is controlled by changing the applied field. The global flow diagram in the two-parameter space is obtained by using a two-parameter position-space renormalization-group approach. The crossover exponent and the crossover radius are calculated. The crossover phenomenon is described in terms of a thermodynamic representation of the two-phase equilibrium.
Barnsley, Michael F
2012-01-01
""Difficult concepts are introduced in a clear fashion with excellent diagrams and graphs."" - Alan E. Wessel, Santa Clara University""The style of writing is technically excellent, informative, and entertaining."" - Robert McCartyThis new edition of a highly successful text constitutes one of the most influential books on fractal geometry. An exploration of the tools, methods, and theory of deterministic geometry, the treatment focuses on how fractal geometry can be used to model real objects in the physical world. Two sixteen-page full-color inserts contain fractal images, and a bonus CD of
Operator spaces and residually finite-dimensional $C^{*}$-algebras
Pestov, V G
1993-01-01
For every operator space $X$ the $C^\\ast$-algebra containing it in a universal way is residually finite-dimensional (that is, has a separating family of finite-dimensional representations). In particular, the free $C^\\ast$-algebra on any normed space so is. This is an extension of an earlier result by Goodearl and Menal, and our short proof is based on a criterion due to Exel and Loring.
RRT+ : Fast Planning for High-Dimensional Configuration Spaces
Xanthidis, Marios; Rekleitis, Ioannis; O'Kane, Jason M.
2016-01-01
In this paper we propose a new family of RRT based algorithms, named RRT+ , that are able to find faster solutions in high-dimensional configuration spaces compared to other existing RRT variants by finding paths in lower dimensional subspaces of the configuration space. The method can be easily applied to complex hyper-redundant systems and can be adapted by other RRT based planners. We introduce RRT+ and develop some variants, called PrioritizedRRT+ , PrioritizedRRT+-Connect, and Prioritize...
Vector cross product in n-dimensional vector space
Tian, Xiu-Lao; Yang, Chao; Hu, Yang; Tian, Chao
2013-01-01
The definition of vector cross product (VCP) introduced by Eckmann only exists in thethree- and the seven- dimensional vector space. In this paper, according to the orthogonal completeness, magnitude of basis vector cross product and all kinds of combinations of basis vector $\\hat{e}_i$, the generalized definition of VCP in the odd n-dimensional vector space is given by introducing a cross term $X_{AB}$. In addition, the definition is validated by reducing the generalization definition to the...
RRT+ : Fast Planning for High-Dimensional Configuration Spaces
Xanthidis, Marios; Rekleitis, Ioannis; O'Kane, Jason M.
2016-01-01
In this paper we propose a new family of RRT based algorithms, named RRT+ , that are able to find faster solutions in high-dimensional configuration spaces compared to other existing RRT variants by finding paths in lower dimensional subspaces of the configuration space. The method can be easily applied to complex hyper-redundant systems and can be adapted by other RRT based planners. We introduce RRT+ and develop some variants, called PrioritizedRRT+ , PrioritizedRRT+-Connect, and Prioritize...
Coset space dimensional reduction of Einstein-Yang-Mills theory
Energy Technology Data Exchange (ETDEWEB)
Chatzistavrakidis, A. [Institute of Nuclear Physics, NCSR Demokritos, 15310 Athens (Greece); Physics Department, National Technical University of Athens, 15780 Zografou Campus, Athens (Greece); Manousselis, P. [Physics Department, National Technical University of Athens, 15780 Zografou Campus, Athens (Greece); Department of Engineering Sciences, University of Patras, 26110 Patras (Greece); Prezas, N. [Theory Unit, Physics Department, 1211 Geneva (Switzerland); Zoupanos, G.
2008-04-15
In the present contribution we extend our previous work by considering the coset space dimensional reduction of higher-dimensional Einstein-Yang-Mills theories including scalar fluctuations as well as Kaluza-Klein excitations of the compactification metric and we describe the gravity-modified rules for the reduction of non-abelian gauge theories. (Abstract Copyright [2008], Wiley Periodicals, Inc.)
IDENTIFICATION OF ARCHITECTURAL FUNCTIONS IN A FOUR-DIMENSIONAL SPACE
Directory of Open Access Journals (Sweden)
Firza Utama Sjarifudin
2012-05-01
Full Text Available This research has explored the possibilities and concept of architectural space in a virtual environment. The virtual environment exists as a different concept, and challenges the constraints of the physical world. One of the possibilities in a virtual environment is that it is able to extend the spatial dimension higher than the physical three-dimension. To take the advantage of this possibility, this research has applied some geometrical four-dimensional (4D methods to define virtual architectural space. The spatial characteristics of 4D space is established by analyzing the four-dimensional structure that can be comprehended by human participant for its spatial quality, and by developing a system to control the fourth axis of movement. Multiple three-dimensional spaces that fluidly change their volume have been defined as one of the possibilities of virtual architectural space concept in order to enrich our understanding of virtual spatial experience.
Energy Technology Data Exchange (ETDEWEB)
Benenti, Giuliano; Casati, Giulio; Guarneri, Italo; Terraneo, Marcello
2001-07-02
We numerically analyze quantum survival probability fluctuations in an open, classically chaotic system. In a quasiclassical regime and in the presence of classical mixed phase space, such fluctuations are believed to exhibit a fractal pattern, on the grounds of semiclassical arguments. In contrast, we work in a classical regime of complete chaoticity and in a deep quantum regime of strong localization. We provide evidence that fluctuations are still fractal, due to the slow, purely quantum algebraic decay in time produced by dynamical localization. Such findings considerably enlarge the scope of the existing theory.
Turbulence on a Fractal Fourier set
Lanotte, Alessandra Sabina; Biferale, Luca; Malapaka, Shiva Kumar; Toschi, Federico
2015-01-01
The dynamical effects of mode reduction in Fourier space for three dimensional turbulent flows is studied. We present fully resolved numerical simulations of the Navier-Stokes equations with Fourier modes constrained to live on a fractal set of dimension D. The robustness of the energy cascade and vortex stretching mechanisms are tested at changing D, from the standard three dimensional case to a strongly decimated case for D = 2.5, where only about $3\\%$ of the Fourier modes interact. While the direct energy cascade persist, deviations from the Kolmogorov scaling are observed in the kinetic energy spectra. A model in terms of a correction with a linear dependency on the co-dimension of the fractal set, $E(k)\\sim k^{- 5/3 + 3 -D }$, explains the results. At small scales, the intermittent behaviour due to the vorticity production is strongly modified by the fractal decimation, leading to an almost Gaussian statistics already at D ~ 2.98. These effects are connected to a genuine modification in the triad-to-tri...
Astaneh, Amin Faraji
2015-01-01
We use the Heat Kernel method to calculate the Entanglement Entropy for a given entangling region on a fractal. The leading divergent term of the entropy is obtained as a function of the fractal dimension as well as the walk dimension. The power of the UV cut-off parameter is (generally) a fractional number which indeed is a certain combination of these two indices. This exponent is known as the spectral dimension. We show that there is a novel log periodic oscillatory behavior in the entropy which has root in the complex dimension of a fractal. We finally indicate that the Holographic calculation in a certain Hyper-scaling violating bulk geometry yields the same leading term for the entanglement entropy, if one identifies the effective dimension of the hyper-scaling violating theory with the spectral dimension of the fractal. We provide more supports with comparing the behavior of the thermal entropy in terms of the temperature in these two cases.
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
The main contents in this note are: 1. introduction; 2. locally compact groups and local fields ; 3. calcaius on fractals based upon local fields; 4. fractional calculus and fractals; 5. fractal function spaces and PDE on fractals.
Iannaccone, Stephen; Zhou, Yue; Walterhouse, David; Taborn, Greg; Landini, Gabriel; Iannaccone, Philip
2012-01-01
The production of organ parenchyma in a rapid and reproducible manner is critical to normal development. In chimeras produced by the combination of genetically distinguishable tissues, mosaic patterns of cells derived from the combined genotypes can be visualized. These patterns comprise patches of contiguously similar genotypes and are different in different organs but similar in a given organ from individual to individual. Thus, the processes that produce the patterns are regulated and conserved. We have previously established that mosaic patches in multiple tissues are fractal, consistent with an iterative, recursive growth model with simple stereotypical division rules. Fractal dimensions of various tissues are consistent with algorithmic models in which changing a single variable (e.g. daughter cell placement after division) switches the mosaic pattern from islands to stripes of cells. Here we show that the spiral pattern previously observed in mouse cornea can also be visualized in rat chimeras. While it is generally held that the pattern is induced by stem cell division dynamics, there is an unexplained discrepancy in the speed of cellular migration and the emergence of the pattern. We demonstrate in chimeric rat corneas both island and striped patterns exist depending on the age of the animal. The patches that comprise the pattern are fractal, and the fractal dimension changes with the age of the animal and indicates the constraint in patch complexity as the spiral pattern emerges. The spiral patterns are consistent with a loxodrome. Such data are likely to be relevant to growth and cell division in organ systems and will help in understanding how organ parenchyma are generated and maintained from multipotent stem cell populations located in specific topographical locations within the organ. Ultimately, understanding algorithmic growth is likely to be essential in achieving organ regeneration in vivo or in vitro from stem cell populations.
Directory of Open Access Journals (Sweden)
Stephen Iannaccone
Full Text Available The production of organ parenchyma in a rapid and reproducible manner is critical to normal development. In chimeras produced by the combination of genetically distinguishable tissues, mosaic patterns of cells derived from the combined genotypes can be visualized. These patterns comprise patches of contiguously similar genotypes and are different in different organs but similar in a given organ from individual to individual. Thus, the processes that produce the patterns are regulated and conserved. We have previously established that mosaic patches in multiple tissues are fractal, consistent with an iterative, recursive growth model with simple stereotypical division rules. Fractal dimensions of various tissues are consistent with algorithmic models in which changing a single variable (e.g. daughter cell placement after division switches the mosaic pattern from islands to stripes of cells. Here we show that the spiral pattern previously observed in mouse cornea can also be visualized in rat chimeras. While it is generally held that the pattern is induced by stem cell division dynamics, there is an unexplained discrepancy in the speed of cellular migration and the emergence of the pattern. We demonstrate in chimeric rat corneas both island and striped patterns exist depending on the age of the animal. The patches that comprise the pattern are fractal, and the fractal dimension changes with the age of the animal and indicates the constraint in patch complexity as the spiral pattern emerges. The spiral patterns are consistent with a loxodrome. Such data are likely to be relevant to growth and cell division in organ systems and will help in understanding how organ parenchyma are generated and maintained from multipotent stem cell populations located in specific topographical locations within the organ. Ultimately, understanding algorithmic growth is likely to be essential in achieving organ regeneration in vivo or in vitro from stem cell populations.
Marks-Tarlow, Terry
Linear concepts of time plus the modern capacity to track history emerged out of circular conceptions characteristic of ancient and traditional cultures. A fractal concept of time lies implicitly within the analog clock, where each moment is treated as unique. With fractal geometry the best descriptor of nature, qualities of self-similarity and scale invariance easily model her endless variety and recursive patterning, both in time and across space. To better manage temporal aspects of our lives, a fractal concept of time is non-reductive, based more on the fullness of being than on fragments of doing. By using a fractal concept of time, each activity or dimension of life is multiply and vertically nested. Each nested cycle remains simultaneously present, operating according to intrinsic dynamics and time scales. By adding the vertical axis of simultaneity to the horizontal axis of length, time is already full and never needs to be filled. To attend to time's vertical dimension is to tap into the imaginary potential for infinite depth. To switch from linear to fractal time allows us to relax into each moment while keeping in mind the whole.
Combinatorial fractal Brownian motion model
Institute of Scientific and Technical Information of China (English)
朱炬波; 梁甸农
2000-01-01
To solve the problem of how to determine the non-scaled interval when processing radar clutter using fractal Brownian motion (FBM) model, a concept of combinatorial FBM model is presented. Since the earth (or sea) surface varies diversely with space, a radar clutter contains several fractal structures, which coexist on all scales. Taking the combination of two FBMs into account, via theoretical derivation we establish a combinatorial FBM model and present a method to estimate its fractal parameters. The correctness of the model and the method is proved by simulation experiments and computation of practial data. Furthermore, we obtain the relationship between fractal parameters when processing combinatorial model with a single FBM model. Meanwhile, by theoretical analysis it is concluded that when combinatorial model is observed on different scales, one of the fractal structures is more obvious.
Baker, Robert G. V.
2017-02-01
Self-similar matrices of the fine structure constant of solar electromagnetic force and its inverse, multiplied by the Carrington synodic rotation, have been previously shown to account for at least 98% of the top one hundred significant frequencies and periodicities observed in the ACRIM composite irradiance satellite measurement and the terrestrial 10.7cm Penticton Adjusted Daily Flux data sets. This self-similarity allows for the development of a time-space differential equation (DE) where the solutions define a solar model for transmissions through the core, radiative, tachocline, convective and coronal zones with some encouraging empirical and theoretical results. The DE assumes a fundamental complex oscillation in the solar core and that time at the tachocline is smeared with real and imaginary constructs. The resulting solutions simulate for tachocline transmission, the solar cycle where time-line trajectories either 'loop' as Hermite polynomials for an active Sun or 'tail' as complementary error functions for a passive Sun. Further, a mechanism that allows for the stable energy transmission through the tachocline is explored and the model predicts the initial exponential coronal heating from nanoflare supercharging. The twisting of the field at the tachocline is then described as a quaternion within which neutrinos can oscillate. The resulting fractal bubbles are simulated as a Julia Set which can then aggregate from nanoflares into solar flares and prominences. Empirical examples demonstrate that time and space fractals are important constructs in understanding the behaviour of the Sun, from the impact on climate and biological histories on Earth, to the fractal influence on the spatial distributions of the solar system. The research suggests that there is a fractal clock underpinning solar frequencies in packages defined by the fine structure constant, where magnetic flipping and irradiance fluctuations at phase changes, have periodically impacted on the
Aspects of the Supersymmetry Algebra in Four Dimensional Euclidean Space
McKeon, D G C
1998-01-01
The simplest supersymmetry (SUSY) algebra in four dimensional Euclidean space ($4dE$) has been shown to closely resemble the $N = 2$ SUSY algebra in four dimensional Minkowski space ($4dM$). The structure of the former algebra is examined in greater detail in this paper. We first present its Clifford algebra structure. This algebra shows that the momentum Casimir invariant of physical states has an upper bound which is fixed by the central charges. Secondly, we use reduction of the $N = 1$ SUSY algebra in six dimensional Minkowski space ($6dM$) to $4dE$; this reproduces our SUSY algebra in $4dE$. Moreover, this same reduction of supersymmetric Yang-Mills theory (SSYM) in $6dM$ reproduces Zumino's SSYM in $4dE$. We demonstrate how this dimensional reduction can be used to introduce additional generators into the SUSY algebra in $4dE$.
Modelling Parsing Constraints with High-Dimensional Context Space.
Burgess, Curt; Lund, Kevin
1997-01-01
Presents a model of high-dimensional context space, the Hyperspace Analogue to Language (HAL), with a series of simulations modelling human empirical results. Proposes that HAL's context space can be used to provide a basic categorization of semantic and grammatical concepts; model certain aspects of morphological ambiguity in verbs; and provide…
Wilson flux breaking and coset space dimensional reduction
Energy Technology Data Exchange (ETDEWEB)
Zoupanos, G.
1988-02-11
Higher dimensional gauge theories lead, after dimensional reduction on coset spaces, to four-dimensional gauge theories usually with the natural emergence of a Higgs sector which is completely determined. However, the Higgs fields never appear in the adjoint representation which in many GUTs could lead to a successful spontaneous symmetry breaking towards the low energy gauge group. As an alternative we suggest that the breaking of the four-dimensional GUTs obtained from CSDR could be provided by the Wilson flux breaking and we discuss some semirealistic examples. We also speculate on the possibility that the breaking of the electroweak sector has dynamical origin.
Some basic inequalities in higher dimensional non-Euclid space
Institute of Scientific and Technical Information of China (English)
2007-01-01
In this paper, the concept of a finite mass-points system∑N(H(A))(N>n) being in a sphere in an n-dimensional hyperbolic space Hn and a finite mass-points system∑N(S(A))(N>n) being in a hyperplane in an n-dimensional spherical space Sn is introduced, then, the rank of the Cayley-Menger matrix AN(H)(or a AN(S)) of the finite mass-points system∑∑N(S(A))(or∑N(S(A))) in an n-dimensional hyperbolic space Hn (or spherical space Sn) is no more than n+2 when∑N(H(A))(N>n) (or∑N(S(A))(N>n)) are in a sphere (or hyperplane). On the one hand, the Yang-Zhang’s inequalities, the Neuberg-Pedoe’s inequalities and the inequality of the metric addition in an n-dimensional hyperbolic space Hn and in an n-dimensional spherical space Sn are established by the method of characteristic roots. These are basic inequalities in hyperbolic geometry and spherical geometry. On the other hand, some relative problems and conjectures are brought.
Execution spaces for simple higher dimensional automata
DEFF Research Database (Denmark)
Raussen, Martin
2012-01-01
of allowable (d-)paths. In Raussen (Algebr Geom Topol 10:1683–1714, 2010), we developed a new method describing, for a certain subclass of HDA, the homotopy type of the space of execution paths (d-paths) as a finite simplicial complex. Several restrictions that were made to ease the presentation in that latter...
Fractal patterns of fractures in granites
Velde, B.; Dubois, J.; Moore, D.; Touchard, G.
1991-01-01
Fractal measurements using the Cantor's dust method in a linear one-dimensional analysis mode were made on the fracture patterns revealed on two-dimensional, planar surfaces in four granites. This method allows one to conclude that: 1. (1)|The fracture systems seen on two-dimensional surfaces in granites are consistent with the part of fractal theory that predicts a repetition of patterns on different scales of observation, self similarity. Fractal analysis gives essentially the same values of D on the scale of kilometres, metres and centimetres (five orders of magnitude) using mapped, surface fracture patterns in a Sierra Nevada granite batholith (Mt. Abbot quadrangle, Calif.). 2. (2)|Fractures show the same fractal values at different depths in a given batholith. Mapped fractures (main stage ore veins) at three mining levels (over a 700 m depth interval) of the Boulder batholith, Butte, Mont. show the same fractal values although the fracture disposition appears to be different at different levels. 3. (3)|Different sets of fracture planes in a granite batholith, Central France, and in experimental deformation can have different fractal values. In these examples shear and tension modes have the same fractal values while compressional fractures follow a different fractal mode of failure. The composite fracture patterns are also fractal but with a different, median, fractal value compared to the individual values for the fracture plane sets. These observations indicate that the fractal method can possibly be used to distinguish fractures of different origins in a complex system. It is concluded that granites fracture in a fractal manner which can be followed at many scales. It appears that fracture planes of different origins can be characterized using linear fractal analysis. ?? 1991.
Oleshko, Klaudia; de Jesús Correa López, María; Romero, Alejandro; Ramírez, Victor; Pérez, Olga
2016-04-01
for the reservoir' hydraulic units distribution in space and time, as well as for the corresponding well testing data. References: 1. Mandelbrot, B., 1995. Foreword to Fractals in Petroleum Geology and Earth Processes, Edited by: Christopher C. Barton and Paul R. La Pointe, Plenum Press, New York: vii-xii. 2. Jin-Zhou Zhao, Cui-Cui Sheng, Yong_Ming Li, and Shun-Chu Li, 2015. A Mathematical Model for the Analysis of the Pressure Transient Response of Fluid Flow in Fractal Reservoir. J. of Chemistry, ID 596597, 8p. 3. Siler, T. , 2007. Fractal Reactor. International Conference Series on Emerging Nuclear Energy Systems 4. Corbett, P. W. M., 2012. The Role of Geoengineering in field development. INTECH, Chapter 8: 181- 198. 5. Nelson, P.H. and J. Kibler, 2003. A Catalog of Porosity and Permeability from core plugs in siliciclastic rocks. U.S. Geological Survey. 6. Per Bak and Kan Chen, 1989. The Physics of Fractals. Physica D 38: 5-12.
Fractals and Scars on a Compact Octagon
Levin, J; Levin, Janna; Barrow, John D.
2000-01-01
A finite universe naturally supports chaotic classical motion. An ordered fractal emerges from the chaotic dynamics which we characterize in full for a compact 2-dimensional octagon. In the classical to quantum transition, the underlying fractal can persist in the form of scars, ridges of enhanced amplitude in the semiclassical wave function. Although the scarring is weak on the octagon, we suggest possible subtle implications of fractals and scars in a finite universe.
Edge effect causes apparent fractal correlation dimension of uniform spatial raindrop distribution
Uijlenhoet, R.; Porra, J.M.; Sempere Torres, D.; Creutin, J.D.
2009-01-01
Lovejoy and Schertzer (1990a) presented a statistical analysis of blotting paper observations of the (two-dimensional) spatial distribution of raindrop stains. They found empirical evidence for the fractal scaling behavior of raindrops in space, with potentially far-reaching implications for rainfal
Some basic inequalities in higher dimensional non-Euclid space
Institute of Scientific and Technical Information of China (English)
Ding-hua YANG
2007-01-01
In this paper, the concept of a finite mass-points system ∑N(H(A))(N ＞ n) being in a sphere in an n-dimensional hyperbolic space Hn and a finite mass-points system ∑N(S(A))(N ＞ n)being in a hyperplane in an n-dimensional spherical space Sn is introduced, then, the rank of the CayleyMenger matrix -ΛN(H) (or a -ΛN(S)) of the finite mass-points system ∑N(S(A)) (or ∑N(S(A))) in an ndimensional hyperbolic space Hn (or spherical space Sn) is no more than n+2 when ∑N(H(A))(N ＞ n)(or ∑N(S(A))(N ＞ n)) are in a sphere (or hyperplane). On the one hand, the Yang-Zhang's inequalities, the Neuberg-Pedoe's inequalities and the inequality of the metric addition in an n-dimensional hyperbolic space Hn and in an n-dimensional spherical space Sn are established by the method of characteristic roots. These are basic inequalities in hyperbolic geometry and spherical geometry. On the other hand, some relative problems and conjectures are brought.
Visualising very large phylogenetic trees in three dimensional hyperbolic space
Directory of Open Access Journals (Sweden)
Liberles David A
2004-04-01
Full Text Available Abstract Background Common existing phylogenetic tree visualisation tools are not able to display readable trees with more than a few thousand nodes. These existing methodologies are based in two dimensional space. Results We introduce the idea of visualising phylogenetic trees in three dimensional hyperbolic space with the Walrus graph visualisation tool and have developed a conversion tool that enables the conversion of standard phylogenetic tree formats to Walrus' format. With Walrus, it becomes possible to visualise and navigate phylogenetic trees with more than 100,000 nodes. Conclusion Walrus enables desktop visualisation of very large phylogenetic trees in 3 dimensional hyperbolic space. This application is potentially useful for visualisation of the tree of life and for functional genomics derivatives, like The Adaptive Evolution Database (TAED.
Networks embedded in n-dimensional space : The impact of dimensionality change
Peli, Gabor; Bruggeman, Jeroen
2006-01-01
Social networks can be embedded in an n-dimensional space, where the dimensions may reveal or denote underlying properties of interest. When the pertaining actors occupy niches of resources in this space, e.g., organizational niches of affiliates, we show there exists a non-monotonic effect of dimen
Autonomous mental development in high dimensional context and action spaces.
Joshi, Ameet; Weng, Juyang
2003-01-01
Autonomous Mental Development (AMD) of robots opened a new paradigm for developing machine intelligence, using neural network type of techniques and it fundamentally changed the way an intelligent machine is developed from manual to autonomous. The work presented here is a part of SAIL (Self-Organizing Autonomous Incremental Learner) project which deals with autonomous development of humanoid robot with vision, audition, manipulation and locomotion. The major issue addressed here is the challenge of high dimensional action space (5-10) in addition to the high dimensional context space (hundreds to thousands and beyond), typically required by an AMD machine. This is the first work that studies a high dimensional (numeric) action space in conjunction with a high dimensional perception (context state) space, under the AMD mode. Two new learning algorithms, Direct Update on Direction Cosines (DUDC) and High-Dimensional Conjugate Gradient Search (HCGS), are developed, implemented and tested. The convergence properties of both the algorithms and their targeted applications are discussed. Autonomous learning of speech production under reinforcement learning is studied as an example.
Mannheim Curves in Nonflat 3-Dimensional Space Forms
Directory of Open Access Journals (Sweden)
Wenjing Zhao
2015-01-01
Full Text Available We consider the Mannheim curves in nonflat 3-dimensional space forms (Riemannian or Lorentzian and we give the concept of Mannheim curves. In addition, we investigate the properties of nonnull Mannheim curves and their partner curves. We come to the conclusion that a necessary and sufficient condition is that a linear relationship with constant coefficients will exist between the curvature and the torsion of the given original curves. In the case of null curve, we reveal that there are no null Mannheim curves in the 3-dimensional de Sitter space.
The use of virtual reality to reimagine two-dimensional representations of three-dimensional spaces
Fath, Elaine
2015-03-01
A familiar realm in the world of two-dimensional art is the craft of taking a flat canvas and creating, through color, size, and perspective, the illusion of a three-dimensional space. Using well-explored tricks of logic and sight, impossible landscapes such as those by surrealists de Chirico or Salvador Dalí seem to be windows into new and incredible spaces which appear to be simultaneously feasible and utterly nonsensical. As real-time 3D imaging becomes increasingly prevalent as an artistic medium, this process takes on an additional layer of depth: no longer is two-dimensional space restricted to strategies of light, color, line and geometry to create the impression of a three-dimensional space. A digital interactive environment is a space laid out in three dimensions, allowing the user to explore impossible environments in a way that feels very real. In this project, surrealist two-dimensional art was researched and reimagined: what would stepping into a de Chirico or a Magritte look and feel like, if the depth and distance created by light and geometry were not simply single-perspective illusions, but fully formed and explorable spaces? 3D environment-building software is allowing us to step into these impossible spaces in ways that 2D representations leave us yearning for. This art project explores what we gain--and what gets left behind--when these impossible spaces become doors, rather than windows. Using sketching, Maya 3D rendering software, and the Unity Engine, surrealist art was reimagined as a fully navigable real-time digital environment. The surrealist movement and its key artists were researched for their use of color, geometry, texture, and space and how these elements contributed to their work as a whole, which often conveys feelings of unexpectedness or uneasiness. The end goal was to preserve these feelings while allowing the viewer to actively engage with the space.
Alfven wave in higher dimensional space time
Panigrahi, D; Chatterjee, S
2009-01-01
Following the wellknown spacetime decomposition technique as applied to (d+1) dimensions we write down the equations of magnetohydrodynamics (MHD) in a spatially at generalised FRW universe. Assuming an equation of state for the background cosmic fluid we find solutions in turn for acous- tic waves and also for Alfven waves in a warm (cold) magnetised plasma. Interestingly the different plasma modes closely resemble the at space coun- terparts except that here the field variables all redshift with their time due to the expansion of the background. It is observed that in the ultrarelativistic limit the field parameters all scale as the free photon. The situation changes in the prerelativistic limit where the frequencies change in a bizarre fashion depending on initial conditions. It is observed that for a fixed magnetic field in a particular medium the Alfven wave velocity decreases with the number of dimensions, being the maximum in the usual 4D. Further for a fixed dimension the velocity attenuation is more ...
Principal Manifolds and Nonlinear Dimensionality Reduction via Tangent Space Alignment
Institute of Scientific and Technical Information of China (English)
张振跃; 查宏远
2004-01-01
We present a new algorithm for manifold learning and nonlinear dimensionality reduction. Based on a set of unorganized da-ta points sampled with noise from a parameterized manifold, the local geometry of the manifold is learned by constructing an approxi-mation for the tangent space at each point, and those tangent spaces are then aligned to give the global coordinates of the data pointswith respect to the underlying manifold. We also present an error analysis of our algorithm showing that reconstruction errors can bequite small in some cases. We illustrate our algorithm using curves and surfaces both in 2D/3D Euclidean spaces and higher dimension-al Euclidean spaces. We also address several theoretical and algorithmic issues for further research and improvements.
Gravitational anomalies in higher dimensional Riemann Cartan space
Yajima, S.; Tokuo, S.; Fukuda, M.; Higashida, Y.; Kamo, Y.; Kubota, S.-I.; Taira, H.
2007-02-01
By applying the covariant Taylor expansion method of the heat kernel, the covariant Einstein anomalies associated with a Weyl fermion of spin \\frac{1}{2} in four-, six- and eight-dimensional Riemann Cartan space are manifestly given. Many unknown terms with torsion tensors appear in these anomalies. The Lorentz anomaly is intimately related to the Einstein anomaly even in Riemann Cartan space. The explicit form of the Lorentz anomaly corresponding to the Einstein anomaly is also obtained.
Gravitational anomalies in higher dimensional Riemann-Cartan space
Energy Technology Data Exchange (ETDEWEB)
Yajima, S [Department of Physics, Kumamoto University, 2-39-1 Kurokami, Kumamoto 860-8555 (Japan); Tokuo, S [Department of Physics, Kumamoto University, 2-39-1 Kurokami, Kumamoto 860-8555 (Japan); Fukuda, M [Department of Physics, Kumamoto University, 2-39-1 Kurokami, Kumamoto 860-8555 (Japan); Higashida, Y [Takuma National College of Technology, 551 kohda, Takuma-cho, Mitoyo, Kagawa 769-1192 (Japan); Kamo, Y [Radioisotope Center, Kyushu University, 3-1-1 Maidashi, Higashi-ku, Fukuoka 812-8582 (Japan); Kubota, S-I [Computing and Communications Center, Kagoshima University, 1-21-35 Koorimoto, Kagoshima 890-0065 (Japan); Taira, H [Department of Physics, Kumamoto University, 2-39-1 Kurokami, Kumamoto 860-8555 (Japan)
2007-02-21
By applying the covariant Taylor expansion method of the heat kernel, the covariant Einstein anomalies associated with a Weyl fermion of spin 1/2 in four-, six- and eight-dimensional Riemann-Cartan space are manifestly given. Many unknown terms with torsion tensors appear in these anomalies. The Lorentz anomaly is intimately related to the Einstein anomaly even in Riemann-Cartan space. The explicit form of the Lorentz anomaly corresponding to the Einstein anomaly is also obtained.
Institute of Scientific and Technical Information of China (English)
黄健; 戴正德
2004-01-01
在本文中,我们在Banach空间考虑二维广义Ginzburg-Landau方程的指数吸引子,且得到其分形维度估计.%In this paper, we consider the exponential attractor for the derivative two - dimensional Ginzburg - Landau equation in Banach space Xαp and also obtain the estimation of the fractal dimension.
Three-Dimensional Space to Assess Cloud Interoperability
2013-03-01
major cloud providers, OpenStack and OpeNebula, to demonstrate the usage of the three-dimensional space and its benefits . We start this chapter with a...documentation:rel4.0:external_auth. [68] X. Gao, P. Shah, A. Yoga , A. Kodgire and X. Ni. Cloud storage survey [Online]. Available: http
Gravitating Instantons In 3 Dimensional Anti de Sitter Space
Ferstl, A; Weir, V; Ferstl, Andrew; Tekin, Bayram; Weir, Victor
2000-01-01
We study the Einstein-Chern-Simons gravity coupled to Yang-Mills-Higgs theoryin three dimensional Euclidean space with cosmological constant. The classicalequations of motion reduce to Bogomol'nyi type first order equations. There areBPS type instanton (monopole) solutions of finite action which we findnumerically. In addition we point out to some exact solutions which aresingular.
Mishra, Jibitesh
2007-01-01
The book covers all the fundamental aspects of generating fractals through L-system. Also it provides insight to various researches in this area for generating fractals through L-system approach & estimating dimensions. Also it discusses various applications of L-system fractals. Key Features: - Fractals generated from L-System including hybrid fractals - Dimension calculation for L-system fractals - Images & codes for L-system fractals - Research directions in the area of L-system fractals - Usage of various freely downloadable tools in this area - Fractals generated from L-System including hybrid fractals- Dimension calculation for L-system fractals- Images & codes for L-system fractals- Research directions in the area of L-system fractals- Usage of various freely downloadable tools in this area
New Fractal Localized Structures in Boiti-Leon-Pempinelli System
Institute of Scientific and Technical Information of China (English)
MAZheng-Yi; ZHUJia-Min; ZHENGChun-Long
2004-01-01
A novel phenomenon that the localized coherent structures of a (2+1)-dimensional physical model possess fractal behaviors is revealed. To clarify the interesting phenomenon, we take the (2+1)-dimensional Boiti Leon-Pempinelli system as a concrete example. Starting from an extended homogeneous balance approach, a general solution of the system is derived. From which some special localized excitations with fractal behaviors are obtained by introducin gsome types of lower-dimensional fractal patterns.
Directory of Open Access Journals (Sweden)
Hai-Feng Zhang
2016-08-01
Full Text Available In this paper, the properties of photonic band gaps (PBGs in two types of two-dimensional plasma-dielectric photonic crystals (2D PPCs under a transverse-magnetic (TM wave are theoretically investigated by a modified plane wave expansion (PWE method where Monte Carlo method is introduced. The proposed PWE method can be used to calculate the band structures of 2D PPCs which possess arbitrary-shaped filler and any lattice. The efficiency and convergence of the present method are discussed by a numerical example. The configuration of 2D PPCs is the square lattices with fractal Sierpinski gasket structure whose constituents are homogeneous and isotropic. The type-1 PPCs is filled with the dielectric cylinders in the plasma background, while its complementary structure is called type-2 PPCs, in which plasma cylinders behave as the fillers in the dielectric background. The calculated results reveal that the enough accuracy and good convergence can be obtained, if the number of random sampling points of Monte Carlo method is large enough. The band structures of two types of PPCs with different fractal orders of Sierpinski gasket structure also are theoretically computed for a comparison. It is demonstrate that the PBGs in higher frequency region are more easily produced in the type-1 PPCs rather than in the type-2 PPCs. Sierpinski gasket structure introduced in the 2D PPCs leads to a larger cutoff frequency, enhances and induces more PBGs in high frequency region. The effects of configurational parameters of two types of PPCs on the PBGs are also investigated in detail. The results show that the PBGs of the PPCs can be easily manipulated by tuning those parameters. The present type-1 PPCs are more suitable to design the tunable compacted devices.
Zhang, Hai-Feng; Liu, Shao-Bin
2016-08-01
In this paper, the properties of photonic band gaps (PBGs) in two types of two-dimensional plasma-dielectric photonic crystals (2D PPCs) under a transverse-magnetic (TM) wave are theoretically investigated by a modified plane wave expansion (PWE) method where Monte Carlo method is introduced. The proposed PWE method can be used to calculate the band structures of 2D PPCs which possess arbitrary-shaped filler and any lattice. The efficiency and convergence of the present method are discussed by a numerical example. The configuration of 2D PPCs is the square lattices with fractal Sierpinski gasket structure whose constituents are homogeneous and isotropic. The type-1 PPCs is filled with the dielectric cylinders in the plasma background, while its complementary structure is called type-2 PPCs, in which plasma cylinders behave as the fillers in the dielectric background. The calculated results reveal that the enough accuracy and good convergence can be obtained, if the number of random sampling points of Monte Carlo method is large enough. The band structures of two types of PPCs with different fractal orders of Sierpinski gasket structure also are theoretically computed for a comparison. It is demonstrate that the PBGs in higher frequency region are more easily produced in the type-1 PPCs rather than in the type-2 PPCs. Sierpinski gasket structure introduced in the 2D PPCs leads to a larger cutoff frequency, enhances and induces more PBGs in high frequency region. The effects of configurational parameters of two types of PPCs on the PBGs are also investigated in detail. The results show that the PBGs of the PPCs can be easily manipulated by tuning those parameters. The present type-1 PPCs are more suitable to design the tunable compacted devices.
Eliazar, Iddo; Klafter, Joseph
2008-06-01
We explore six classes of fractal probability laws defined on the positive half-line: Weibull, Frechét, Lévy, hyper Pareto, hyper beta, and hyper shot noise. Each of these classes admits a unique statistical power-law structure, and is uniquely associated with a certain operation of renormalization. All six classes turn out to be one-dimensional projections of underlying Poisson processes which, in turn, are the unique fixed points of Poissonian renormalizations. The first three classes correspond to linear Poissonian renormalizations and are intimately related to extreme value theory (Weibull, Frechét) and to the central limit theorem (Lévy). The other three classes correspond to nonlinear Poissonian renormalizations. Pareto's law--commonly perceived as the "universal fractal probability distribution"--is merely a special case of the hyper Pareto class.
Fractal structure and fractal dimension determination at nanometer scale
Institute of Scientific and Technical Information of China (English)
张跃; 李启楷; 褚武扬; 王琛; 白春礼
1999-01-01
Three-dimensional fractures of different fractal dimensions have been constructed with successive random addition algorithm, the applicability of various dimension determination methods at nanometer scale has been studied. As to the metallic fractures, owing to the limited number of slit islands in a slit plane or limited datum number at nanometer scale, it is difficult to use the area-perimeter method or power spectrum method to determine the fractal dimension. Simulation indicates that box-counting method can be used to determine the fractal dimension at nanometer scale. The dimensions of fractures of valve steel 5Cr21Mn9Ni4N have been determined with STM. Results confirmed that fractal dimension varies with direction at nanometer scale. Our study revealed that, as to theoretical profiles, the dependence of fractal dimension with direction is simply owing to the limited data set number, i.e. the effect of boundaries. However, the dependence of fractal dimension with direction at nanometer scale in rea
Dimensional reduction of ten-dimensional E/sub 8/ gauge theory over a compact coset space S/R
Energy Technology Data Exchange (ETDEWEB)
Luest, D.; Zoupanos, G.
1985-12-26
Dimensional reduction of pure gauge theories over a compact coset space S/R leads to four-dimensional Yang-Mills-Higgs theories. We present a complete analysis of the four-dimensional unified models obtained by dimensionally reducing an E/sub 8/ gauge theory in ten dimensions over all possible six-dimensional homogeneous spaces S/R when S is a subgroup of E/sub 8/ and simple. (orig.).
Signal propagation in 2-dimensional threshold cellular space.
Kobuchi, Y
1976-11-25
A two-dimensional network of uniformly connected McCulloch-Pitts neurons is considered and signal propagations in the network are analyzed. The problems are set up in the framework of cellular space such that each cell is a copy of any given McCulloch-Pitts neuron and is connected to the nearest neighboring cells. It is assumed that the threshold value is positive and that there exists only one firing cell at the beginning. Then it is shown that essentially there are only four signal propagation patterns and a firing pattern at any time t can be obtained by such a superposition of the propagation patterns that includes the newly defined concept of dominance and assimilation. The exact formulae representing firing patterns at any time t are obtained for any finite rectangle cell space with constant 0 (i.e., non-firing) boundary condition and for the entire two-dimensional cellular space.
Rodríguez, Javier; Prieto, Signed; Ortiz, Liliana; Correa, Catalina; Wiesner,Carolina; Díaz, Martha
2010-01-01
Introducción. La geometría fractal ha mostrado ser adecuada en la descripción matemática de objetos irregulares; esta medida se ha denominado dimensión fractal. La aplicación del análisis fractal para medir los contornos de las células normales así como aquellas que presentan algún tipo de anormalidad, ha mostrado la posibilidad de caracterización matemática de su irregularidad. Objetivos. Medir, a partir de la geometría fractal células del epitelio escamoso de cuello uterino clasificadas com...
Lung cancer-a fractal viewpoint.
Lennon, Frances E; Cianci, Gianguido C; Cipriani, Nicole A; Hensing, Thomas A; Zhang, Hannah J; Chen, Chin-Tu; Murgu, Septimiu D; Vokes, Everett E; Vannier, Michael W; Salgia, Ravi
2015-11-01
Fractals are mathematical constructs that show self-similarity over a range of scales and non-integer (fractal) dimensions. Owing to these properties, fractal geometry can be used to efficiently estimate the geometrical complexity, and the irregularity of shapes and patterns observed in lung tumour growth (over space or time), whereas the use of traditional Euclidean geometry in such calculations is more challenging. The application of fractal analysis in biomedical imaging and time series has shown considerable promise for measuring processes as varied as heart and respiratory rates, neuronal cell characterization, and vascular development. Despite the advantages of fractal mathematics and numerous studies demonstrating its applicability to lung cancer research, many researchers and clinicians remain unaware of its potential. Therefore, this Review aims to introduce the fundamental basis of fractals and to illustrate how analysis of fractal dimension (FD) and associated measurements, such as lacunarity (texture) can be performed. We describe the fractal nature of the lung and explain why this organ is particularly suited to fractal analysis. Studies that have used fractal analyses to quantify changes in nuclear and chromatin FD in primary and metastatic tumour cells, and clinical imaging studies that correlated changes in the FD of tumours on CT and/or PET images with tumour growth and treatment responses are reviewed. Moreover, the potential use of these techniques in the diagnosis and therapeutic management of lung cancer are discussed.
On Differential Equations Describing 3-Dimensional Hyperbolic Spaces
Institute of Scientific and Technical Information of China (English)
WU Jun-Yi; DING Qing; Keti Tenenblat
2006-01-01
In this paper, we introduce the notion of a (2+1)-dimensional differential equation describing threedimensional hyperbolic spaces (3-h.s.). The (2+1)-dimensional coupled nonlinear Schrodinger equation and its sister equation, the (2+1)-dimensional coupled derivative nonlinear Schrodinger equation, are shown to describe 3-h.s. The (2+1)-dimensional generalized HF model: St = (1/2i [S, Sy] + 2iσS)x, σx = 1-4itr(SSxSy), in which S ∈ GLC(2)/GLC(1)×GLC(1),provides another example of (2+1)-dimensional differential equations describing 3-h.s. As a direct consequence, the geometric construction of an infinite number of conservation laws of such equations is illustrated. Furthermore we display a new infinite number of conservation laws of the (2+1)-dimensional nonlinear Schrodinger equation and the (2+1)odimensional derivative nonlinear Schrodinger equation by a geometric way.
BIVARIATE FRACTAL INTERPOLATION FUNCTIONS ON RECTANGULAR DOMAINS
Institute of Scientific and Technical Information of China (English)
Xiao-yuan Qian
2002-01-01
Non-tensor product bivariate fractal interpolation functions defined on gridded rectangular domains are constructed. Linear spaces consisting of these functions are introduced.The relevant Lagrange interpolation problem is discussed. A negative result about the existence of affine fractal interpolation functions defined on such domains is obtained.
Kukushkin, A. B.; Rantsev-Kartinov, V. A.
2003-10-01
The phenomenon of skeletal structures (tubules, cartwheels, and their simple combinations) formerly found in laboratory plasmas, is extended to atmospheric and cosmic phenomena (Phys. Lett. A 306, 175, Dec. 2002). The long-lived filaments in plasmas have been suggested (1998) to possess a skeleton self-assembled during electric breakdown from wildly produced nanodust. The proof-of-concept studies revealed the skeletal structures in (1) plasmas in tokamaks, Z-pinches, plasma focus (in the range 0.01-10 cm), including the electric breakdown stage of discharge, (2) dust deposits in tokamak (10 nm - 10 microns), (3) hailstones (1-10 cm), tornado (10 m -1 km), (4) a wide class of objects in space (10^11-10^23 cm). The similarity of, and the trend toward self-similarity in, skeletal structures suggest all them to possess a fractal condensed matter of particular topology of the fractal. Here we discuss probable role of skeletal structures in the fast nonlocal transport of energy in strongly localized severe weather phenomena (tornado) and laboratory plasmas.
Clustering and dimensionality reduction for image retrieval in high-dimensional spaces
Directory of Open Access Journals (Sweden)
Soumia Benkrama
2014-12-01
Full Text Available The scalability of indexing techniques and image retrieval pose many problems. Indeed, their performance degrades rapidly when the database size increases. In this paper, we propose an efficient indexing method for high-dimensional spaces. We investigate how high-dimensional indexing methods can be used on a partitioned space into clusters to help the design of an efficient and robust CBIR scheme. We develop a new method for efficient clustering is used for structuring objects in the feature space; this method allows dividing the base into data groups according to their similarity, in function of the parameter threshold and vocabulary size. A comparative study is presented between the proposed method and a set of classification methods. The experiments results on the Pascal Visual Object Classes challenges (VOC of 2007 and Caltech-256 dataset show that our method significantly improves the performance. Experimental retrieval results based on the precision/recall measures show interesting results.
Comparison of two fractal interpolation methods
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
Some Properties of Fractals Generated by Linear Cellular Automata
Institute of Scientific and Technical Information of China (English)
倪天佳
2003-01-01
Fractals and cellular automata are both significant areas of research in nonlinear analysis. This paper studies a class of fractals generated by cellular automata. The patterns produced by cellular automata give a special sequence of sets in Euclidean space. The corresponding limit set is shown to be a fractal and the dimension is independent of the choice of the finite initial seed. As opposed to previous works, the fractals here do not depend on the time parameter.
Introducing the Dimensional Continuous Space-Time Theory
Martini, Luiz Cesar
2013-04-01
This article is an introduction to a new theory. The name of the theory is justified by the dimensional description of the continuous space-time of the matter, energy and empty space, that gathers all the real things that exists in the universe. The theory presents itself as the consolidation of the classical, quantum and relativity theories. A basic equation that describes the formation of the Universe, relating time, space, matter, energy and movement, is deduced. The four fundamentals physics constants, light speed in empty space, gravitational constant, Boltzmann's constant and Planck's constant and also the fundamentals particles mass, the electrical charges, the energies, the empty space and time are also obtained from this basic equation. This theory provides a new vision of the Big-Bang and how the galaxies, stars, black holes and planets were formed. Based on it, is possible to have a perfect comprehension of the duality between wave-particle, which is an intrinsic characteristic of the matter and energy. It will be possible to comprehend the formation of orbitals and get the equationing of atomics orbits. It presents a singular comprehension of the mass relativity, length and time. It is demonstrated that the continuous space-time is tridimensional, inelastic and temporally instantaneous, eliminating the possibility of spatial fold, slot space, worm hole, time travels and parallel universes. It is shown that many concepts, like dark matter and strong forces, that hypothetically keep the cohesion of the atomics nucleons, are without sense.
Four-dimensional space groups for pedestrians: composite structures.
Sun, Junliang; Lee, Stephen; Lin, Jianhua
2007-10-01
Higher-dimensional crystals have been studied for the last thirty years. However, most practicing chemists, materials scientists, and crystallographers continue to eschew the use of higher-dimensional crystallography in their work. Yet it has become increasingly clear in recent years that the number of higher-dimensional systems continues to grow from hundreds to as many as a thousand different compounds. Part of the problem has to do with the somewhat opaque language that has developed over the past decades to describe higher-dimensional systems. This language, while well-suited to the specialist, is too sophisticated for the neophyte wishing to enter the field, and as such can be an impediment. This Focus Review hopes to address this issue. The goal of this article is to show the regular chemist or materials scientist that knowledge of regular 3D crystallography is all that is really necessary to understand 4D crystal systems. To this end, we have couched higher-dimensional composite structures in the language of ordinary 3D crystals. In particular, we developed the principle of complementarity, which allows one to identify correctly 4D space groups solely from examination of the two 3D components that make up a typical 4D composite structure.
Kinks in two-dimensional Anti-de Sitter Space
Barnes, J L; ter Veldhuis, T; Webster, M J
2009-01-01
Soliton solutions in scalar field theory defined on a two-dimensional Anti-de Sitter background space-time are investigated. It is shown that the lowest soliton excitation generically has frequency equal to the inverse radius of the space-time. Analytic and numerical soliton solutions are determined in "phi to the fourth" scalar field theory with a negative mass-squared. The classical soliton mass is calculated as a function of the ratio of the square of the mass scale of the field theory over the curvature of the space-time. For the case that this ratio equals unity, the soliton excitation spectrum is determined algebraically and the one-loop radiative correction to the soliton mass is computed in the semi-classical approximation.
Finite element contact analysis of fractal surfaces
Energy Technology Data Exchange (ETDEWEB)
Sahoo, Prasanta; Ghosh, Niloy [Department of Mechanical Engineering, Jadavpur University, Kolkata 700032 (India)
2007-07-21
The present study considers finite element analysis of non-adhesive, frictionless elastic/elastic-plastic contact between a rigid flat plane and a self-affine fractal rough surface using the commercial finite element package ANSYS. Three-dimensional rough surfaces are generated using a modified two-variable Weierstrass-Mandelbrot function with given fractal parameters. Parametric studies are done to consider the general relations between contact properties and key material and surface parameters. The present analysis is validated with available experimental results in the literature. Non-dimensional contact area and displacement are obtained as functions of non-dimensional load for varying fractal surface parameters in the case of elastic contact and for varying rates of strain hardening in the case of elastic-plastic contact of fractal surfaces.
Moskvin, P. P.; Krizhanovskii, V. B.; Rashkovetskii, L. V.; Vuichik, N. V.
2016-05-01
Multifractal (MF) analysis is used to describe the volume of space forms on the surfaces of structures in the solid solution of a Zn x Cd1- x Te-Si(111) substrate. AFM images of film surfaces have been are used for MF analysis. The parameters of MF spectra are determined for the distribution of volume of surface nanoforms. Based on the formal approach and data on the parameters of the fractal state for the volume and surfaces of nanoforms, an equation is proposed that considers the contribution from the fractal structure of the surface to its surface energy. The behavior of the system's surface energy, depending on fractal parameters that describe states of the volume and surfaces of nanoforms is discussed.
Automated photogrammetry for three-dimensional models of urban spaces
Leberl, Franz; Meixner, Philipp; Wendel, Andreas; Irschara, Arnold
2012-02-01
The location-aware Internet is inspiring intensive work addressing the automated assembly of three-dimensional models of urban spaces with their buildings, circulation spaces, vegetation, signs, even their above-ground and underground utility lines. Two-dimensional geographic information systems (GISs) and municipal utility information exist and can serve to guide the creation of models being built with aerial, sometimes satellite imagery, streetside images, indoor imaging, and alternatively with light detection and ranging systems (LiDARs) carried on airplanes, cars, or mounted on tripods. We review the results of current research to automate the information extraction from sensor data. We show that aerial photography at ground sampling distances (GSD) of 1 to 10 cm is well suited to provide geometry data about building facades and roofs, that streetside imagery at 0.5 to 2 cm is particularly interesting when it is collected within community photo collections (CPCs) by the general public, and that the transition to digital imaging has opened the no-cost option of highly overlapping images in support of a more complete and thus more economical automation. LiDAR-systems are a widely used source of three-dimensional data, but they deliver information not really superior to digital photography.
Fractal profit landscape of the stock market.
Grönlund, Andreas; Yi, Il Gu; Kim, Beom Jun
2012-01-01
We investigate the structure of the profit landscape obtained from the most basic, fluctuation based, trading strategy applied for the daily stock price data. The strategy is parameterized by only two variables, p and q Stocks are sold and bought if the log return is bigger than p and less than -q, respectively. Repetition of this simple strategy for a long time gives the profit defined in the underlying two-dimensional parameter space of p and q. It is revealed that the local maxima in the profit landscape are spread in the form of a fractal structure. The fractal structure implies that successful strategies are not localized to any region of the profit landscape and are neither spaced evenly throughout the profit landscape, which makes the optimization notoriously hard and hypersensitive for partial or limited information. The concrete implication of this property is demonstrated by showing that optimization of one stock for future values or other stocks renders worse profit than a strategy that ignores fluctuations, i.e., a long-term buy-and-hold strategy.
Fractal profit landscape of the stock market.
Directory of Open Access Journals (Sweden)
Andreas Grönlund
Full Text Available We investigate the structure of the profit landscape obtained from the most basic, fluctuation based, trading strategy applied for the daily stock price data. The strategy is parameterized by only two variables, p and q Stocks are sold and bought if the log return is bigger than p and less than -q, respectively. Repetition of this simple strategy for a long time gives the profit defined in the underlying two-dimensional parameter space of p and q. It is revealed that the local maxima in the profit landscape are spread in the form of a fractal structure. The fractal structure implies that successful strategies are not localized to any region of the profit landscape and are neither spaced evenly throughout the profit landscape, which makes the optimization notoriously hard and hypersensitive for partial or limited information. The concrete implication of this property is demonstrated by showing that optimization of one stock for future values or other stocks renders worse profit than a strategy that ignores fluctuations, i.e., a long-term buy-and-hold strategy.
Pavlov, Y V
2001-01-01
One derived expressions for the vacuum mean values of energy-momentum tensor of the scalar field with arbitrary relation to curvature in N-dimensional quasi-euclidean space-time for vacuum. One generalized n-wave procedure for multidimensional spaces. One calculated all counter-members for N=5 and for a conformal scalar field in N=6, 7. One determined the geometric structure of three first counter-members for N-dimensional spaces. All subtractions in 4-dimensional space-time and 3 first subtractions in multidimensional spaces are shown to correspond to renormalization of constants of priming and gravitational Lagrangian
Lévy processes on a generalized fractal comb
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.
Image Encryption using chaos functions and fractal key
Directory of Open Access Journals (Sweden)
Houman Kashanian
2016-09-01
Full Text Available Many image in recent years are transmitted via internet and stored on it. Maintain the confidentiality of these data has become a major issue. So that encryption algorithms permit only authorized users to access data which is a proper solution to this problem.This paper presents a novel scheme for image encryption. At first, a two dimensional logistic mapping is applied to permutation relations between image pixels. We used a fractal image as an encryption key. Given that the chaotic mapping properties such as extreme sensitivity to initial values, random behavior, non-periodic, certainty and so on, we used theses mappings in order to select fractal key for encryption. Experimental results show that proposed algorithm to encrypt image has many features. Due to features such as large space key, low relations between the pixels of encrypted image, high sensitivity to key and high security, it can effectively protect the encrypted image security.
Transient chaos and fractal structures in planetary feeding zones
Kovács, Tamás
2014-01-01
The circular restricted three body problem is investigated in the context of accretion and scattering processes. In our model a large number of identical non-interacting mass-less planetesimals are considered in planar case orbiting a star-planet system. This description allows us to investigate in dynamical systems approach the gravitational scattering and possible captures of the particles by the forming planetary embryo. Although the problem serves a large variety of complex motion, the results can be easily interpreted because of the low dimensionality of the phase space. We show that initial conditions define isolated regions of the disk, where accretion or escape of the planetesimals occur, these have, in fact, a fractal structure. The fractal geometry of these "basins" implies that the dynamics is very complex. Based on the calculated escape rates and escape times, it is also demonstrated that the planetary accretion rate is exponential for short times and follows a power-law for longer integration. A ...
Wang, Wenlong; Moore, M. A.; Katzgraber, Helmut G.
2017-09-01
The fractal dimension of excitations in glassy systems gives information on the critical dimension at which the droplet picture of spin glasses changes to a description based on replica symmetry breaking where the interfaces are space filling. Here, the fractal dimension of domain-wall interfaces is studied using the strong-disorder renormalization group method pioneered by Monthus [Fractals 23, 1550042 (2015), 10.1142/S0218348X15500425] both for the Edwards-Anderson spin-glass model in up to 8 space dimensions, as well as for the one-dimensional long-ranged Ising spin-glass with power-law interactions. Analyzing the fractal dimension of domain walls, we find that replica symmetry is broken in high-enough space dimensions. Because our results for high-dimensional hypercubic lattices are limited by their small size, we have also studied the behavior of the one-dimensional long-range Ising spin-glass with power-law interactions. For the regime where the power of the decay of the spin-spin interactions with their separation distance corresponds to 6 and higher effective space dimensions, we find again the broken replica symmetry result of space filling excitations. This is not the case for smaller effective space dimensions. These results show that the dimensionality of the spin glass determines which theoretical description is appropriate. Our results will also be of relevance to the Gardner transition of structural glasses.
The Electron in Three-Dimensional Momentum Space
Mantovani, L.; Bacchetta, A.; Pasquini, B.
2016-07-01
We study the electron as a system composed of an electron and a photon and derive the leading-twist transverse-momentum-dependent distribution functions for both the electron and photon in the dressed electron, thereby offering a three-dimensional description of the dressed electron in momentum space. To obtain the distribution functions, we apply both the formalism of light-front wave function overlap representation and the diagrammatic approach; we discuss the comparison of our results between light-cone gauge and Feynman gauge, discussing the role of the Wilson lines to obtain gauge-independent results. We provide examples of plots of the computed distributions.
Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms
Institute of Scientific and Technical Information of China (English)
2008-01-01
Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in S4, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them the interesting duality theorem holds. As an application spacelike Willmore 2-spheres are classified. Finally we construct a family of homogeneous spacelike Willmore tori.
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Hollerbach, K.; Van Vorhis, R.L. [Lawrence Livermore National Lab., CA (United States); Hollister, A. [Louisiana State Univ., Shreveport, LA (United States)
1996-03-01
Wrist posture and rapid wrist movements are risk factors for work related musculoskeletal disorders. Measurement studies frequently involve optoelectronic methods in which markers are placed on the subject`s hand and wrist and the trajectories of the markers are tracked in three dimensional space. A goal of wrist posture measurements is to quantitatively establish wrist posture orientation. Accuracy and fidelity of the measurement data with respect to kinematic mechanisms are essential in wrist motion studies. Fidelity with the physical kinematic mechanism can be limited by the choice of kinematic modeling techniques and the representation of motion. Frequently, ergonomic studies involving wrist kinematics make use of two dimensional measurement and analysis techniques. Two dimensional measurement of human joint motion involves the analysis of three dimensional displacements in an obersver selected measurement plane. Accurate marker placement and alignment of joint motion plane with the observer plane are difficult. In nature, joint axes can exist at any orientation and location relative to an arbitrarily chosen global reference frame. An arbitrary axis is any axis that is not coincident with a reference coordinate. We calculate the errors that result from measuring joint motion about an arbitrary axis using two dimensional methods.
Finite-dimensional Hilbert space and frame quantization
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Cotfas, Nicolae [Faculty of Physics, University of Bucharest, PO Box 76-54, Post Office 76, Bucharest (Romania); Gazeau, Jean Pierre [Laboratoire APC, Universite Paris Diderot, 10, rue A Domon et L Duquet, 75205 Paris Cedex 13 (France); Vourdas, Apostol, E-mail: ncotfas@yahoo.com, E-mail: gazeau@apc.univ-paris7.fr, E-mail: A.Vourdas@bradford.ac.uk [Department of Computing, University of Bradford, Bradford BD7 1DP (United Kingdom)
2011-04-29
The quantum observables used in the case of quantum systems with finite-dimensional Hilbert space are defined either algebraically in terms of an orthonormal basis and discrete Fourier transformation or by using a continuous system of coherent states. We present an alternative approach to these important quantum systems based on the finite frame quantization. Finite systems of coherent states, usually called finite tight frames, can be defined in a natural way in the case of finite quantum systems. Novel examples of such tight frames are presented. The quantum observables used in our approach are obtained by starting from certain classical observables described by functions defined on the discrete phase space corresponding to the system. They are obtained by using a finite frame and a Klauder-Berezin-Toeplitz-type quantization. Semi-classical aspects of tight frames are studied through lower symbols of basic classical observables.
Random diffusion and cooperation in continuous two-dimensional space.
Antonioni, Alberto; Tomassini, Marco; Buesser, Pierre
2014-03-07
This work presents a systematic study of population games of the Prisoner's Dilemma, Hawk-Dove, and Stag Hunt types in two-dimensional Euclidean space under two-person, one-shot game-theoretic interactions, and in the presence of agent random mobility. The goal is to investigate whether cooperation can evolve and be stable when agents can move randomly in continuous space. When the agents all have the same constant velocity cooperation may evolve if the agents update their strategies imitating the most successful neighbor. If a fitness difference proportional is used instead, cooperation does not improve with respect to the static random geometric graph case. When viscosity effects set-in and agent velocity becomes a quickly decreasing function of the number of neighbors they have, one observes the formation of monomorphic stable clusters of cooperators or defectors in the Prisoner's Dilemma. However, cooperation does not spread in the population as in the constant velocity case.
Marks-Tarlow, Terry
2010-01-01
In this article, the author draws on contemporary science to illuminate the relationship between early play experiences, processes of self-development, and the later emergence of the fractal self. She argues that orientation within social space is a primary function of early play and developmentally a two-step process. With other people and with…
Time evolution of quantum fractals
Wojcik; Bialynicki-Birula; Zyczkowski
2000-12-11
We propose a general construction of wave functions of arbitrary prescribed fractal dimension, for a wide class of quantum problems, including the infinite potential well, harmonic oscillator, linear potential, and free particle. The box-counting dimension of the probability density P(t)(x) = |Psi(x,t)|(2) is shown not to change during the time evolution. We prove a universal relation D(t) = 1+Dx/2 linking the dimensions of space cross sections Dx and time cross sections D(t) of the fractal quantum carpets.
Time Evolution of Quantum Fractals
Wójcik, D; Zyczkowski, K; Wojcik, Daniel; Bialynicki-Birula, Iwo; Zyczkowski, Karol
2000-01-01
We propose a general construction of wave functions of arbitrary prescribed fractal dimension, for a wide class of quantum problems, including the infinite potential well, harmonic oscillator, linear potential and free particle. The box-counting dimension of the probability density $P_t(x)=|\\Psi(x,t)|^2$ is shown not to change during the time evolution. We prove a universal relation $D_t=1+D_x/2$ linking the dimensions of space cross-sections $D_x$ and time cross-sections $D_t$ of the fractal quantum carpets.
Life and space dimensionality: A brief review of old and new entangled arguments
Caruso, Francisco
2016-01-01
A general sketch on how the problem of space dimensionality depends on anthropic arguments is presented. Several examples of how life has been used to constraint space dimensionality (and vice-versa) are reviewed. In particular, the influences of three-dimensionality in the solar system stability and the origin of life on Earth are discussed. New constraints on space dimensionality and on its invariance in very large spatial and temporal scales are also stressed.
Space-Time--Time Five-dimensional Kaluza--Weyl Space
Ellis, H G
2001-01-01
Space-time--time couples Kaluza's five-dimensional geometry with Weyl's conformal space-time geometry to produce an extension that goes beyond what either of those theories can achieve by itself. Kaluza's ``cylinder condition'' is replaced by an ``exponential expansion constraint'' that causes translations along the secondary time dimension to induce both the electromagnetic gauge transformations found in the Kaluza and the Weyl theories and the metrical gauge transformations unique to the Weyl theory, related as Weyl had postulated. A space-time--time geodesic describes a test particle whose rest mass, space-time momentum, and electric charge q, all defined kinematically, evolve in accord with definite dynamical laws. Its motion is governed by four apparent forces: the Einstein gravitational force, the Lorentz electromagnetic force, a force proportional to the electromagnetic potential, and a force proportional to a scalar field's gradient d(ln phi). The test particles exhibit quantum behavior: (1) they appe...
Reconstructing the fractal dimension of granular aggregates from light intensity spectra.
Tang, Fiona H M; Maggi, Federico
2015-12-21
There has been growing interest in using the fractal dimension to study the hierarchical structures of soft materials after realising that fractality is an important property of natural and engineered materials. This work presents a method to quantify the internal architecture and the space-filling capacity of granular fractal aggregates by reconstructing the three-dimensional capacity dimension from their two-dimensional optical projections. Use is made of the light intensity of the two-dimensional aggregate images to describe the aggregate surface asperities (quantified by the perimeter-based fractal dimension) and the internal architecture (quantified by the capacity dimension) within a mathematical framework. This method was tested on control aggregates of diffusion-limited (DLA), cluster-cluster (CCA) and self-correlated (SCA) types, stereolithographically-fabricated aggregates, and experimentally-acquired natural sedimentary aggregates. Statistics of the reconstructed capacity dimension featured correlation coefficients R ≥ 98%, residuals NRMSE ≤ 10% and percent errors PE ≤ 4% as compared to controls, and improved earlier approaches by up to 50%.
Directory of Open Access Journals (Sweden)
Kendra A Batchelder
Full Text Available 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.
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.
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Naumis, Gerardo G. [Departamento de Fisica-Qimica, Instituto de Fisica, Universidad Nacional Autonoma de Mexico, Apartado Postal 20-364, 01000 Mexico, Distrito Federal (Mexico); Facultad de Ciencias, Universidad Autonoma del Estado de Morelos, Av. Universidad 10001, 62210 Cuernavaca, Morelos (Mexico); Departamento de Fisica-Matematica, Universidad Iberoamericana, Prolongacion Paseo de la Reforma 880, Col. Lomas de Santa Fe. 01210, Mexico DF (Mexico)], E-mail: naumis@fisica.unam.mx; Bazan, A. [Departamento de Fisica-Qimica, Instituto de Fisica, Universidad Nacional Autonoma de Mexico, Apartado Postal 20-364, 01000 Mexico, Distrito Federal (Mexico); Facultad de Ciencias, Universidad Autonoma del Estado de Morelos, Av. Universidad 10001, 62210 Cuernavaca, Morelos (Mexico); Departamento de Fisica-Matematica, Universidad Iberoamericana, Prolongacion Paseo de la Reforma 880, Col. Lomas de Santa Fe. 01210, Mexico DF (Mexico); Torres, M. [Instituto de Fisica Aplicada, Consejo Superior de Investigaciones Cientifica, Serrano 144, 28006 Madrid (Spain); Aragon, J.L.; Quintero-Torres, R. [Departamento de Nanotecnologia, Centro de Fisica Aplicada y Tecnologia Avanzada, Universidad Nacional Autonoma de Mexico, Apartado Postal 1-1010, 76000 Queretaro (Mexico)
2008-09-01
One of the few examples in which the physical properties of an incommensurable system reflect an underlying higher dimensionality is presented. Specifically, we show that the reflectivity distribution of an incommensurable one-dimensional cavity is given by the density of states of a tight-binding Hamiltonian in a two-dimensional triangular lattice. Such effect is due to an independent phase decoupling of the scattered waves, produced by the incommensurable nature of the system, which mimics a random noise generator. This principle can be applied to design a cavity that avoids resonant reflections for almost any incident wave. An optical analogy, by using three mirrors with incommensurable distances between them, is also presented. Such array produces a countable infinite fractal set of reflections, a phenomena which is opposite to the effect of optical invisibility.
Fernández-Guasti, M.
The quadratic iteration is mapped within a nondistributive imaginary scator algebra in 1 + 2 dimensions. The Mandelbrot set is identically reproduced at two perpendicular planes where only the scalar and one of the hypercomplex scator director components are present. However, the bound three-dimensional S set projections change dramatically even for very small departures from zero of the second hypercomplex plane. The S set exhibits a rich fractal-like boundary in three dimensions. Periodic points with period m, are shown to be necessarily surrounded by points that produce a divergent magnitude after m iterations. The scator set comprises square nilpotent elements that ineluctably belong to the bound set. Points that are square nilpotent on the mth iteration, have preperiod 1 and period m. Two-dimensional plots are presented to show some of the main features of the set. A three-dimensional rendering reveals the highly complex structure of its boundary.
Fat fractal percolation and k-fractal percolation
Broman, Erik; Camia, Federico; Joosten, Matthijs; Meester, Ronald
2011-01-01
We consider two variations on the Mandelbrot fractal percolation model. In the k-fractal percolation model, the d-dimensional unit cube is divided in N^d equal subcubes, k of which are retained while the others are discarded. The procedure is then iterated inside the retained cubes at all smaller scales. We show that the (properly rescaled) percolation critical value of this model converges to the critical value of site percolation in L^d as N tends to infinity. This is analogous to the result of Falconer and Grimmett that the critical value for Mandelbrot fractal percolation converges to the critical value of site percolation in L^d. In the fat fractal percolation model, subcubes are retained with probability p_n at step n of the construction, where (p_n) is a non-decreasing sequence with \\prod p_n > 0. The Lebesgue measure of the limit set is positive a.s. given non-extinction. We show that with probability 1 either the set of "dust" points or the set of connected components larger than one point has positi...
Chromatic $k$-Mean Clustering in High Dimensional Space
Ding, Hu
2012-01-01
In this paper, we study a new type of clustering problem, called {\\em Chromatic Clustering}, in high dimensional space. Chromatic clustering seeks to partition a set of colored points into groups (or clusters) so that no group contains points with the same color and a certain objective function is optimized. In this paper, we focus on $k$-mean clustering, and investigate its hardness and approximation solutions. The additional coloring requirement destroys some key properties used in existing $k$-mean clustering techniques (for the ordinary clustering problem), and significantly complicates the problem. There is no FPTAS for the chromatic clustering problem, even if $k=2$. Based on several new geometric observations and an interesting sphere peeling approach, we show that a near linear time (on $n$ and $d$) $(1+\\epsilon)$-approximation is, however, still achievable for the chromatic clustering problem.
DNA Origami with Complex Curvatures in Three-Dimensional Space
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Han, Dongran; Pal, Suchetan; Nangreave, Jeanette; Deng, Zhengtao; Liu, Yan; Yan, Hao
2011-04-14
We present a strategy to design and construct self-assembling DNA nanostructures that define intricate curved surfaces in three-dimensional (3D) space using the DNA origami folding technique. Double-helical DNA is bent to follow the rounded contours of the target object, and potential strand crossovers are subsequently identified. Concentric rings of DNA are used to generate in-plane curvature, constrained to 2D by rationally designed geometries and crossover networks. Out-of-plane curvature is introduced by adjusting the particular position and pattern of crossovers between adjacent DNA double helices, whose conformation often deviates from the natural, B-form twist density. A series of DNA nanostructures with high curvature—such as 2D arrangements of concentric rings and 3D spherical shells, ellipsoidal shells, and a nanoflask—were assembled.
Packing hyperspheres in high-dimensional Euclidean spaces.
Skoge, Monica; Donev, Aleksandar; Stillinger, Frank H; Torquato, Salvatore
2006-10-01
We present a study of disordered jammed hard-sphere packings in four-, five-, and six-dimensional Euclidean spaces. Using a collision-driven packing generation algorithm, we obtain the first estimates for the packing fractions of the maximally random jammed (MRJ) states for space dimensions d=4, 5, and 6 to be phi(MRJ) approximately 0.46, 0.31, and 0.20, respectively. To a good approximation, the MRJ density obeys the scaling form phi(MRJ)=c1/2(d)+(c2d)/2d, where c1=-2.72 and c2=2.56, which appears to be consistent with the high-dimensional asymptotic limit, albeit with different coefficients. Calculations of the pair correlation function g2(r) and structure factor S(k) for these states show that short-range ordering appreciably decreases with increasing dimension, consistent with a recently proposed "decorrelation principle," which, among other things, states that unconstrained correlations diminish as the dimension increases and vanish entirely in the limit d-->infinity. As in three dimensions (where phi(MRJ) approximately 0.64), the packings show no signs of crystallization, are isostatic, and have a power-law divergence in g2(r) at contact with power-law exponent approximately 0.4. Across dimensions, the cumulative number of neighbors equals the kissing number of the conjectured densest packing close to where g2(r) has its first minimum. Additionally, we obtain estimates for the freezing and melting packing fractions for the equilibrium hard-sphere fluid-solid transition, phi(F) approximately 0.32 and phi(M) approximately 0.39, respectively, for d=4, and phi(F) approximately 0.20 and phi(M) approximately 0.25, respectively, for d=5. Although our results indicate the stable phase at high density is a crystalline solid, nucleation appears to be strongly suppressed with increasing dimension.
On the d-dimensional Quasi-Equally Spaced Sampling
Nordio, Alessandro; Viterbo, Emanuele
2008-01-01
We study a class of random matrices that appear in several communication and signal processing applications, and whose asymptotic eigenvalue distribution is closely related to the reconstruction error of an irregularly sampled bandlimited signal. We focus on the case where the random variables characterizing these matrices are d-dimensional vectors, independent, and quasi-equally spaced, i.e., they have an arbitrary distribution and their averages are vertices of a d-dimensional grid. Although a closed form expression of the eigenvalue distribution is still unknown, under these conditions we are able (i) to derive the distribution moments as the matrix size grows to infinity, while its aspect ratio is kept constant, and (ii) to show that the eigenvalue distribution tends to the Marcenko-Pastur law as d->infinity. These results can find application in several fields, as an example we show how they can be used for the estimation of the mean square error provided by linear reconstruction techniques.
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
Fractal Aggregation Under Rotation
Institute of Scientific and Technical Information of China (English)
WU Feng-Min; WU Li-Li; LU Hang-Jun; LI Qiao-Wen; YE Gao-Xiang
2004-01-01
By means of the Monte Carlo simulation, a fractal growth model is introduced to describe diffusion-limited aggregation (DLA) under rotation. Patterns which are different from the classical DLA model are observed and the fractal dimension of such clusters is calculated. It is found that the pattern of the clusters and their fractal dimension depend strongly on the rotation velocity of the diffusing particle. Our results indicate the transition from fractal to non-fractal behavior of growing cluster with increasing rotation velocity, i.e. for small enough angular velocity ω the fractal dimension decreases with increasing ω, but then, with increasing rotation velocity, the fractal dimension increases and the cluster becomes compact and tends to non-fractal.
Thamrin, Cindy; Stern, Georgette; Frey, Urs
2010-06-01
There is increasing interest in the study of fractals in medicine. In this review, we provide an overview of fractals, of techniques available to describe fractals in physiological data, and we propose some reasons why a physician might benefit from an understanding of fractals and fractal analysis, with an emphasis on paediatric respiratory medicine where possible. Among these reasons are the ubiquity of fractal organisation in nature and in the body, and how changes in this organisation over the lifespan provide insight into development and senescence. Fractal properties have also been shown to be altered in disease and even to predict the risk of worsening of disease. Finally, implications of a fractal organisation include robustness to errors during development, ability to adapt to surroundings, and the restoration of such organisation as targets for intervention and treatment.
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.
Ji-Huan He
2011-01-01
A new fractal derive is defined, which is very easy for engineering applications to discontinuous problems, two simple examples are given to elucidate to establish governing equations with fractal derive and how to solve such equations, respectively.
Fractal Characterization of Hyperspectral Imagery
Qiu, Hon-Iie; Lam, Nina Siu-Ngan; Quattrochi, Dale A.; Gamon, John A.
1999-01-01
Two Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) hyperspectral images selected from the Los Angeles area, one representing urban and the other, rural, were used to examine their spatial complexity across their entire spectrum of the remote sensing data. Using the ICAMS (Image Characterization And Modeling System) software, we computed the fractal dimension values via the isarithm and triangular prism methods for all 224 bands in the two AVIRIS scenes. The resultant fractal dimensions reflect changes in image complexity across the spectral range of the hyperspectral images. Both the isarithm and triangular prism methods detect unusually high D values on the spectral bands that fall within the atmospheric absorption and scattering zones where signature to noise ratios are low. Fractal dimensions for the urban area resulted in higher values than for the rural landscape, and the differences between the resulting D values are more distinct in the visible bands. The triangular prism method is sensitive to a few random speckles in the images, leading to a lower dimensionality. On the contrary, the isarithm method will ignore the speckles and focus on the major variation dominating the surface, thus resulting in a higher dimension. It is seen where the fractal curves plotted for the entire bandwidth range of the hyperspectral images could be used to distinguish landscape types as well as for screening noisy bands.
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.…
Algebraic surface grid generation in three-dimensional space
Warsi, Saif
1992-01-01
An interactive program for algebraic generation of structured surface grids in three dimensional space was developed on the IRIS4D series workstations. Interactive tools are available to ease construction of edge curves and surfaces in 3-D space. Addition, removal, or redistribution of points at arbitrary locations on a general 3-D surface or curve is possible. Also, redistribution of surface grid points may be accomplished through use of conventional surface splines or a method called 'surface constrained transfinite interpolation'. This method allows the user to redistribute the grid points on the edges of a surface patch; the effect of the redistribution is then propagated to the remainder of the surface through a transfinite interpolation procedure where the grid points will be constrained to lie on the surface. The program was written to be highly functional and easy to use. A host of utilities are available to ease the grid generation process. Generality of the program allows the creation of single and multizonal surface grids according to the user requirements. The program communicates with the user through popup menus, windows, and the mouse.
On Space Efficient Two Dimensional Range Minimum Data Structures
DEFF Research Database (Denmark)
Brodal, Gerth Stølting; Davoodi, Pooya; Rao, S. Srinivasa
2012-01-01
the space and query time of the problem. We show that every algorithm enabled to access A during the query and using a data structure of size O(N/c) bits requires Ω(c) query time, for any c where 1≤c≤N. This lower bound holds for arrays of any dimension. In particular, for the one dimensional version...... of the problem, the lower bound is tight up to a constant factor. In two dimensions, we complement the lower bound with an indexing data structure of size O(N/c) bits which can be preprocessed in O(N) time to support O(clog 2 c) query time. For c=O(1), this is the first O(1) query time algorithm using a data...... structure of optimal size O(N) bits. For the case where queries can not probe A, we give a data structure of size O(N⋅min {m,log n}) bits with O(1) query time, assuming m≤n. This leaves a gap to the space lower bound of Ω(Nlog m) bits for this version of the problem...
Model Based Defect Detection in Multi-Dimensional Vector Spaces
Honda, Toshifumi; Obara, Kenji; Harada, Minoru; Igarashi, Hajime
A highly sensitive inspection algorithm is proposed that extracts defects in multidimensional vector spaces from multiple images. The proposed algorithm projects subtraction vectors calculated from test and reference images to control the noise by reducing the dimensionality of vector spaces. The linear projection vectors are optimized using a physical defect model, and the noise distribution is calculated from the images. Because the noise distribution varies with the intensity or texture of the pixels, the target image is divided into small regions and the noise distribution of the subtraction images are calculated for each divided region. The bidirectional local perturbation pattern matching (BD-LPPM) which is an enhanced version of the LPPM, is proposed to increase the sensitivity when calculating the subtraction vectors, especially when the reference image contains more high-frequency components than the test image. The proposed algorithm is evaluated using defect samples for three different scanning electron microscopy images. The results reveal that the proposed algorithm increases the signal-to-noise ratio by a factor of 1.32 relative to that obtained using the Mahalanobis distance algorithm.
Three-Dimensional Quantification of Pore Space in Flocculated Sediments
Lawrence, Tom; Spencer, Kate; Bushby, Andy; Manning, Andrew
2017-04-01
Flocculated sediment structure plays a vital role in determining sediment dynamics within the water column in fresh and saline water bodies. The porosity of flocs contributes to their specific density and therefore their settling characteristics, and can also affect settling characteristics via through-flow. The process of settling and resuspension of flocculated material causes the formation of larger and more complex individual flocs, about which little is known quantitatively of the internal micro-structure and therefore porosity. Hydrological and sedimentological modelling software currently uses estimations of porosity, because it is difficult to capture and analyse flocs. To combat this, we use a novel microscopy method usually performed on biological material to scan the flocs, the output of which can be used to quantify the dimensions and arrangement of pores. This involves capturing flocculated sediment, staining the sample with heavy metal elements to highlight organic content in the Scanning Electron Microscope later, and finally setting the sample in resin. The overall research aim is to quantitatively characterise the dimensions and distribution of pore space in flocs in three dimensions. In order to gather data, Scanning Electron Microscopy and micro-Computed Tomography have been utilised to produce the necessary images to identify and quantify the pore space. The first objective is to determine the dimensional limits of pores in the structure (i.e. what area do they encapsulate? Are they interconnected or discreet?). This requires a repeatable definition to be established, so that all floc pore spaces can be quantified using the same parameters. The LabSFLOC settling column and dyes will be used as one possible method of determining the outer limits of the discreet pore space. LabSFLOC is a sediment settling column that uses a camera to record the flocs, enabling analysis of settling characteristics. The second objective is to develop a reliable
Fractal Reconnection in Solar and Stellar Environments
Shibata, Kazunari
2016-01-01
Recent space based observations of the Sun revealed that magnetic reconnection is ubiquitous in the solar atmosphere, ranging from small scale reconnection (observed as nanoflares) to large scale one (observed as long duration flares or giant arcades). Often the magnetic reconnection events are associated with mass ejections or jets, which seem to be closely related to multiple plasmoid ejections from fractal current sheet. The bursty radio and hard X-ray emissions from flares also suggest the fractal reconnection and associated particle acceleration. We shall discuss recent observations and theories related to the plasmoid-induced-reconnection and the fractal reconnection in solar flares, and their implication to reconnection physics and particle acceleration. Recent findings of many superflares on solar type stars that has extended the applicability of the fractal reconnection model of solar flares to much a wider parameter space suitable for stellar flares are also discussed.
Fractal Fluctuations and Statistical Normal Distribution
Selvam, A M
2008-01-01
Dynamical systems in nature exhibit selfsimilar fractal fluctuations and the corresponding power spectra follow inverse power law form signifying long-range space-time correlations identified as self-organized criticality. The physics of self-organized criticality is not yet identified. The Gaussian probability distribution used widely for analysis and description of large data sets underestimates the probabilities of occurrence of extreme events such as stock market crashes, earthquakes, heavy rainfall, etc. The assumptions underlying the normal distribution such as fixed mean and standard deviation, independence of data, are not valid for real world fractal data sets exhibiting a scale-free power law distribution with fat tails. A general systems theory for fractals visualizes the emergence of successively larger scale fluctuations to result from the space-time integration of enclosed smaller scale fluctuations. The model predicts a universal inverse power law incorporating the golden mean for fractal fluct...
The Representation of Three-Dimensional Space in Fish.
Burt de Perera, Theresa; Holbrook, Robert I; Davis, Victoria
2016-01-01
In mammals, the so-called "seat of the cognitive map" is located in place cells within the hippocampus. Recent work suggests that the shape of place cell fields might be defined by the animals' natural movement; in rats the fields appear to be laterally compressed (meaning that the spatial map of the animal is more highly resolved in the horizontal dimensions than in the vertical), whereas the place cell fields of bats are statistically spherical (which should result in a spatial map that is equally resolved in all three dimensions). It follows that navigational error should be equal in the horizontal and vertical dimensions in animals that travel freely through volumes, whereas in surface-bound animals would demonstrate greater vertical error. Here, we describe behavioral experiments on pelagic fish in which we investigated the way that fish encode three-dimensional space and we make inferences about the underlying processing. Our work suggests that fish, like mammals, have a higher order representation of space that assembles incoming sensory information into a neural unit that can be used to determine position and heading in three-dimensions. Further, our results are consistent with this representation being encoded isotropically, as would be expected for animals that move freely through volumes. Definitive evidence for spherical place fields in fish will not only reveal the neural correlates of space to be a deep seated vertebrate trait, but will also help address the questions of the degree to which environment spatial ecology has shaped cognitive processes and their underlying neural mechanisms.
The representation of three-dimensional space in fish
Directory of Open Access Journals (Sweden)
Theresa eBurt De Perera
2016-03-01
Full Text Available In mammals, the so-called seat of the cognitive map is located in place cells within the hippocampus. Recent work suggests that the shape of place cell fields might be defined by the animals’ natural movement; in rats the fields appear to be laterally compressed (meaning that the spatial map of the animal is more highly resolved in the horizontal dimensions than in the vertical, whereas the place cell fields of bats are statistically spherical (which should result in a spatial map that is equally resolved in all three dimensions. It follows that navigational error should be equal in the horizontal and vertical dimensions in animals that travel freely through volumes, whereas in surface-bound animals would demonstrate greater vertical error. Here, we describe behavioural experiments on pelagic fish in which we investigated the way that fish encode three-dimensional space and we make inferences about the underlying processing. Our work suggests that fish, like mammals, have a higher order representation of space that assembles incoming sensory information into a neural unit that can be used to determine position and heading in three-dimensions. Further, our results are consistent with this representation being encoded isotropically, as would be expected for animals that move freely through volumes. Definitive evidence for spherical place fields in fish will not only reveal the neural correlates of space to be a deep seated vertebrate trait, but will also help address the questions of the degree to which environment spatial ecology has shaped cognitive processes and their underlying neural mechanisms.
Discrete coherent states and probability distributions in finite-dimensional spaces
Energy Technology Data Exchange (ETDEWEB)
Galetti, D.; Marchiolli, M.A.
1995-06-01
Operator bases are discussed in connection with the construction of phase space representatives of operators in finite-dimensional spaces and their properties are presented. It is also shown how these operator bases allow for the construction of a finite harmonic oscillator-like coherent state. Creation and annihilation operators for the Fock finite-dimensional space are discussed and their expressions in terms of the operator bases are explicitly written. The relevant finite-dimensional probability distributions are obtained and their limiting behavior for an infinite-dimensional space are calculated which agree with the well know results. (author). 20 refs, 2 figs.
Wicks, Keith R
1991-01-01
Addressed to all readers with an interest in fractals, hyperspaces, fixed-point theory, tilings and nonstandard analysis, this book presents its subject in an original and accessible way complete with many figures. The first part of the book develops certain hyperspace theory concerning the Hausdorff metric and the Vietoris topology, as a foundation for what follows on self-similarity and fractality. A major feature is that nonstandard analysis is used to obtain new proofs of some known results much more slickly than before. The theory of J.E. Hutchinson's invariant sets (sets composed of smaller images of themselves) is developed, with a study of when such a set is tiled by its images and a classification of many invariant sets as either regular or residual. The last and most original part of the book introduces the notion of a "view" as part of a framework for studying the structure of sets within a given space. This leads to new, elegant concepts (defined purely topologically) of self-similarity and fracta...
Directory of Open Access Journals (Sweden)
FELICIA RAMONA BIRAU
2012-05-01
Full Text Available In this article, the concept of capital market is analysed using Fractal Market Hypothesis which is a modern, complex and unconventional alternative to classical finance methods. Fractal Market Hypothesis is in sharp opposition to Efficient Market Hypothesis and it explores the application of chaos theory and fractal geometry to finance. Fractal Market Hypothesis is based on certain assumption. Thus, it is emphasized that investors did not react immediately to the information they receive and of course, the manner in which they interpret that information may be different. Also, Fractal Market Hypothesis refers to the way that liquidity and investment horizons influence the behaviour of financial investors.
On Materiality and Dimensionality of the Space. Is There Some Unit of the Field?
Directory of Open Access Journals (Sweden)
Belyakov A. V.
2014-10-01
Full Text Available The article presents arguments with a view to recognize that space is material and has possibly a fractal dimension in the range of from three to two. It is proposed that along to the unit of substance (atom Some Unit of the field (vortex tubes should be set. It is shown that the formation of the field structures being a kind “ doubles” of atomic ones is possible. The existence of the three-zone electron structure is confirmed. It is indicated that this concept have already resulted in to the successful explanation of phenomena and to finding of their important parameters at different levels of the organization of matter.
Fractal Aggregation Under Rotation
Institute of Scientific and Technical Information of China (English)
WUFeng-Min; WULi-Li; LUHang-Jun; LIQiao-Wen; YEGao-Xiang
2004-01-01
By means of the Monte Carlo simulation, a fractal growth model is introduced to describe diffusion-limited aggregation (DLA) under rotation. Patterns which are different from the classical DLA model are observed and the fractal dimension of such clusters is calculated. It is found that the pattern of the clusters and their fractal dimension depend strongly on the rotation velocity of the diffusing particle. Our results indicate the transition from fractal to non-fractal behavior of growing cluster with increasing rotation velocity, i.e. for small enough angular velocity ω; thefractal dimension decreases with increasing ω;, but then, with increasing rotation velocity, the fractal dimension increases and the cluster becomes compact and tends to non-fractal.
The Gompertzian curve reveals fractal properties of tumor growth
Energy Technology Data Exchange (ETDEWEB)
Waliszewski, Przemyslaw; Konarski, Jerzy
2003-06-01
The normalized Gompertzian curve reflecting growth of experimental malignant tumors in time can be fitted by the power function y(t)=at{sup b} with the coefficient of nonlinear regression r{>=}0.95, in which the exponent b is a temporal fractal dimension, (i.e., a real number), and time t is a scalar. This curve is a fractal, (i.e., fractal dimension b exists, it changes along the time scale, the Gompertzian function is a contractable mapping of the Banach space R of the real numbers, holds the Banach theorem about the fix point, and its derivative is {<=}1). This denotes that not only space occupied by the interacting cancer cells, but also local, intrasystemic time, in which tumor growth occurs, possesses fractal structure. The value of the mean temporal fractal dimension decreases along the curve approaching eventually integer values; a fact consistent with our hypothesis that the fractal structure is lost during tumor progression.
Energy Technology Data Exchange (ETDEWEB)
Douzas, George; Grammatikopoulos, Theodoros; Zoupanos, George [National Technical University, Physics Department, Athens (Greece)
2009-02-15
We consider a N=1 supersymmetric E{sub 8} gauge theory, defined in ten dimensions and we determine all four-dimensional gauge theories resulting from the generalized dimensional reduction a la Forgacs-Manton over coset spaces, followed by a subsequent application of the Wilson flux spontaneous symmetry-breaking mechanism. Our investigation is constrained only by the requirements that (i) the dimensional reduction leads to the potentially phenomenologically interesting, anomaly-free, four-dimensional E{sub 6}, SO{sub 10} and SU{sub 5} GUTs and (ii) the Wilson flux mechanism makes use only of the freely acting discrete symmetries of all possible six-dimensional coset spaces. (orig.)
Fractal dimension analysis of complexity in Ligeti piano pieces
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.
The literary uses of high-dimensional space
Directory of Open Access Journals (Sweden)
Ted Underwood
2015-12-01
Full Text Available Debates over “Big Data” shed more heat than light in the humanities, because the term ascribes new importance to statistical methods without explaining how those methods have changed. What we badly need instead is a conversation about the substantive innovations that have made statistical modeling useful for disciplines where, in the past, it truly wasn’t. These innovations are partly technical, but more fundamentally expressed in what Leo Breiman calls a new “culture” of statistical modeling. Where 20th-century methods often required humanists to squeeze our unstructured texts, sounds, or images into some special-purpose data model, new methods can handle unstructured evidence more directly by modeling it in a high-dimensional space. This opens a range of research opportunities that humanists have barely begun to discuss. To date, topic modeling has received most attention, but in the long run, supervised predictive models may be even more important. I sketch their potential by describing how Jordan Sellers and I have begun to model poetic distinction in the long 19th century—revealing an arc of gradual change much longer than received literary histories would lead us to expect.
Clinical refraction in three-dimensional dioptric space revisited.
Raasch, T
1997-06-01
The traditional clinical designation of spherocylindrical power unambiguously specifies the refractive properties of a thin lens or refractive surface. This representation of dioptric power is not, however, optimum in mathematical terms, as is apparent when, for example, two spherocylindrical lens powers are added. Alternative systems have been described which are not subject to this same type of difficulty, and the essential feature of these other systems is that spherocylindrical power is defined in terms of a three-dimensional dioptric space in which the axes are usually orthogonal. The advantages of this orthogonality can be exploited in the practice of clinical refraction, provided lens powers in these three dimensions can be physically implemented. Systems using these characteristics have been introduced in the past, but the clinical community has not adopted them on a widespread basis. However, systems which take advantages of these features do have unique advantages relative to traditional clinical refraction procedures. These characteristics, and refractive procedures which exploit their advantages, are described.
Likelihood ratio based verification in high dimensional spaces
Hendrikse, Anne; Veldhuis, Raymond; Spreeuwers, Luuk
2013-01-01
The increase of the dimensionality of data sets often lead to problems during estimation, which are denoted as the curse of dimensionality. One of the problems of Second Order Statistics (SOS) estimation in high dimensional data is that the resulting covariance matrices are not full rank, so their i
Likelihood ratio based verification in high dimensional spaces
Hendrikse, A.J.; Veldhuis, Raymond N.J.; Spreeuwers, Lieuwe Jan
The increase of the dimensionality of data sets often lead to problems during estimation, which are denoted as the curse of dimensionality. One of the problems of Second Order Statistics (SOS) estimation in high dimensional data is that the resulting covariance matrices are not full rank, so their
Rheological and fractal hydrodynamics of aerobic granules.
Tijani, H I; Abdullah, N; Yuzir, A; Ujang, Zaini
2015-06-01
The structural and hydrodynamic features for granules were characterized using settling experiments, predefined mathematical simulations and ImageJ-particle analyses. This study describes the rheological characterization of these biologically immobilized aggregates under non-Newtonian flows. The second order dimensional analysis defined as D2=1.795 for native clusters and D2=1.099 for dewatered clusters and a characteristic three-dimensional fractal dimension of 2.46 depicts that these relatively porous and differentially permeable fractals had a structural configuration in close proximity with that described for a compact sphere formed via cluster-cluster aggregation. The three-dimensional fractal dimension calculated via settling-fractal correlation, U∝l(D) to characterize immobilized granules validates the quantitative measurements used for describing its structural integrity and aggregate complexity. These results suggest that scaling relationships based on fractal geometry are vital for quantifying the effects of different laminar conditions on the aggregates' morphology and characteristics such as density, porosity, and projected surface area.
Two-dimensional imaginary lobachevsky space. Separation of variables and contractions
Energy Technology Data Exchange (ETDEWEB)
Pogosyan, G. S., E-mail: pogosyan@theor.jinr.ru; Yakhno, A. [Universidad de Guadalajara, Departamento de Matematicas, CUCEI (Mexico)
2011-07-15
The Inoenue-Wigner contraction from the SO(2, 1) group to the E(1, 1) group is used to relate the separation of variables in Laplace-Beltrami (Helmholtz) equations for the corresponding two-dimensional homogeneous spaces: two-dimensional one sheeted hyperboloid and two-dimensional pseudo-Euclidean space. Here we consider the contraction limits of some basis functions for the subgroup coordinates only.
Energy Spectrum of Helium Confined to a Two-Dimensional Space
Institute of Scientific and Technical Information of China (English)
XIEWen-Fang
2005-01-01
Making use of the adiabatic hyperspherical approach, we report a calculation for the energy spectrum of the ground and low-excited states of a two-dimensional helium in a magnetic field. The results show that the ground and low-excited states of helium in low-dimensional space are more stable than those in three-dimensional space and there may exist more bound states.
Two-dimensional relativistic space charge limited current flow in the drift space
Energy Technology Data Exchange (ETDEWEB)
Liu, Y. L.; Chen, S. H., E-mail: chensh@ncu.edu.tw [Department of Physics, National Central University, Jhongli 32001, Taiwan (China); Koh, W. S. [A-STAR Institute of High Performance Computing, Singapore 138632 (Singapore); Ang, L. K. [Engineering Product Development, Singapore University of Technology and Design, Singapore 138682 (Singapore)
2014-04-15
Relativistic two-dimensional (2D) electrostatic (ES) formulations have been derived for studying the steady-state space charge limited (SCL) current flow of a finite width W in a drift space with a gap distance D. The theoretical analyses show that the 2D SCL current density in terms of the 1D SCL current density monotonically increases with D/W, and the theory recovers the 1D classical Child-Langmuir law in the drift space under the approximation of uniform charge density in the transverse direction. A 2D static model has also been constructed to study the dynamical behaviors of the current flow with current density exceeding the SCL current density, and the static theory for evaluating the transmitted current fraction and minimum potential position have been verified by using 2D ES particle-in-cell simulation. The results show the 2D SCL current density is mainly determined by the geometrical effects, but the dynamical behaviors of the current flow are mainly determined by the relativistic effect at the current density exceeding the SCL current density.
Cohomology spaces of low dimensional complex associative algebras
Mohammed, Nadia F.; Rakhimov, Isamiddin S.; Hussain, Sharifah Kartini Said
2017-04-01
In this paper, we calculate cohomology groups of low-dimensional complex associative algebras. The calculations are based on a classification result and description of derivations of low-dimensional associative algebras obtained earlier. For the first cohomology group, we give basic cocycles up to inner derivations. We also provide basic coboundaries for the second cohomology groups for low-dimensional associative algebras (including both unital and non unital).
Detecting dimensional crossover and finite Hilbert space through entanglement entropies
Garagiola, Mariano; Cuestas, Eloisa; Pont, Federico M.; Serra, Pablo; Osenda, Omar
2016-01-01
The information content of the two-particle one- and two-dimensional Calogero model is studied using the von Neumann and R\\'enyi entropies. The one-dimensional model is shown to have non-monotonic entropies with finite values in the large interaction strength limit. On the other hand, the von Neumann entropy of the two-dimensional model with isotropic confinement is a monotone increasing function of the interaction strength which diverges logarithmically. By considering an anisotropic confine...
Low dimensionality semiconductors: modelling of excitons via a fractional-dimensional space
Christol, P.; Lefebvre, P.; Mathieu, H.
1993-09-01
An interaction space with a fractionnal dimension is used to calculate in a simple way the binding energies of excitons confined in quantum wells, superlattices and quantum well wires. A very simple formulation provides this energy versus the non-integer dimensionality of the physical environment of the electron-hole pair. The problem then comes to determining the dimensionality α. We show that the latter can be expressed from the characteristics of the microstructure. α continuously varies from 3 (bulk material) to 2 for quantum wells and superlattices, and from 3 to 1 for quantum well wires. Quite a fair agreement is obtained with other theoretical calculations and experimental data, and this model coherently describes both three-dimensional limiting cases for quantum wells (L_wrightarrow 0 and L_wrightarrow infty) and the whole range of periods of the superlattice. Such a simple model presents a great interest for spectroscopists though it does not aim to compete with accurate but often tedious variational calculations. Nous utilisons un espace des interactions doté d'une dimension fractionnaire pour calculer simplement l'énergie de liaison des excitons confinés dans les puits quantiques, superréseaux et fils quantiques. Une formulation très simple donne cette énergie en fonction de la dimensionalité non-entière de l'environnement physique de la paire électron-trou. Le problème revient alors à déterminer cette dimensionalité α, dont nous montrons qu'une expression peut être déduite des caractéristiques de la microstructure. α varie continûment de 3 (matériau massif) à 2 pour un puits quantique ou un superréseau, et de 3 à 1 pour un fil quantique, selon le confinement du mouvement des porteurs. Les comparaisons avec d'autres calculs théoriques et données expérimentales sont toujours très convenables, et cette théorie décrit d'une façon cohérente les limites tridimensionnelles du puits quantique (L_wrightarrow 0 et L
Applications of Fractal Signals
Directory of Open Access Journals (Sweden)
Ion TUTĂNESCU
2008-05-01
Full Text Available "Fractal" term - which in Latin languagedefines something fragmented anomalous - wasintroduced in mathematics by B. B. Mandelbrot in1975. He avoided to define it rigorously and used it todesignate some "rugged" and "self-similar"geometrical forms. Fractals were involved in the theoryof chaotic dynamic systems and used to designatecertain specific sets; finally, they were “captured” bygeometry and remarked in tackling of the boundaryproblems. It proved that the fractals can be of interesteven in the signal’s theory.
Fractal Geometry of Architecture
Lorenz, Wolfgang E.
In Fractals smaller parts and the whole are linked together. Fractals are self-similar, as those parts are, at least approximately, scaled-down copies of the rough whole. In architecture, such a concept has also been known for a long time. Not only architects of the twentieth century called for an overall idea that is mirrored in every single detail, but also Gothic cathedrals and Indian temples offer self-similarity. This study mainly focuses upon the question whether this concept of self-similarity makes architecture with fractal properties more diverse and interesting than Euclidean Modern architecture. The first part gives an introduction and explains Fractal properties in various natural and architectural objects, presenting the underlying structure by computer programmed renderings. In this connection, differences between the fractal, architectural concept and true, mathematical Fractals are worked out to become aware of limits. This is the basis for dealing with the problem whether fractal-like architecture, particularly facades, can be measured so that different designs can be compared with each other under the aspect of fractal properties. Finally the usability of the Box-Counting Method, an easy-to-use measurement method of Fractal Dimension is analyzed with regard to architecture.
Change detection for hyperspectral sensing in a transformed low-dimensional space
Energy Technology Data Exchange (ETDEWEB)
Foy, Bernard R [Los Alamos National Laboratory; Theiler, James [Los Alamos National Laboratory
2010-01-01
We present an approach to the problem of change in hyperspectral imagery that operates in a two-dimensional space. The coordinates in the space are related to Mahalanobis distances for the combined ('stacked') data and the individual hyperspectral scenes. Although it is only two-dimensional, this space is rich enough to include several well-known change detection algorithms, including the hyperbolic anomalous change detector, based on Gaussian scene clutter, and the EC-uncorrelated detector based on heavy-tailed (elliptically contoured) clutter. Because this space is only two-dimensional, adaptive machine learning methods can produce new change detectors without being stymied by the curse of dimensionality. We investigate, in particular, the utility of the support vector machine for learning boundaries in this 2-D space, and compare the performance of the resulting nonlinearly adaptjve detector to change detectors that have themselves shown good performance.
Fractal dimension and architecture of trabecular bone.
Fazzalari, N L; Parkinson, I H
1996-01-01
The fractal dimension of trabecular bone was determined for biopsies from the proximal femur of 25 subjects undergoing hip arthroplasty. The average age was 67.7 years. A binary profile of the trabecular bone in the biopsy was obtained from a digitized image. A program written for the Quantimet 520 performed the fractal analysis. The fractal dimension was calculated for each specimen, using boxes whose sides ranged from 65 to 1000 microns in length. The mean fractal dimension for the 25 subjects was 1.195 +/- 0.064 and shows that in Euclidean terms the surface extent of trabecular bone is indeterminate. The Quantimet 520 was also used to perform bone histomorphometric measurements. These were bone volume/total volume (BV/TV) (per cent) = 11.05 +/- 4.38, bone surface/total volume (BS/TV) (mm2/mm3) = 1.90 +/- 0.51, trabecular thickness (Tb.Th) (mm) = 0.12 +/- 0.03, trabecular spacing (Tb.Sp) (mm) = 1.03 +/- 0.36, and trabecular number (Tb.N) (number/mm) = 0.95 +/- 0.25. Pearsons' correlation coefficients showed a statistically significant relationship between the fractal dimension and all the histomorphometric parameters, with BV/TV (r = 0.85, P fractal dimension shows that trabecular bone exhibits fractal properties over a defined box size, which is within the dimensions of a structural unit for trabecular bone. Therefore, the fractal dimension of trabecular bone provides a measure which does not rely on Euclidean descriptors in order to describe a complex geometry.
Tracking with particle filter for high-dimensional observation and state spaces
Dubuisson, Séverine
2015-01-01
This title concerns the use of a particle filter framework to track objects defined in high-dimensional state-spaces using high-dimensional observation spaces. Current tracking applications require us to consider complex models for objects (articulated objects, multiple objects, multiple fragments, etc.) as well as multiple kinds of information (multiple cameras, multiple modalities, etc.). This book presents some recent research that considers the main bottleneck of particle filtering frameworks (high dimensional state spaces) for tracking in such difficult conditions.
"General theory of a particle mechanics" arising from a fractal surface
Yefremov, Alexander P
2016-01-01
The logical line is traced of formulation of theory of mechanics founded on the basic correlations of mathematics of hypercomplex numbers and associated geometric images. Namely, it is shown that the physical equations of quantum, classical and relativistic mechanics can be regarded as mathematical consequences of a single condition of stability of exceptional algebras of real, complex and quaternion numbers under transformations of primitive constituents of their units and elements. In the course of the study a notion of basic fractal surface underlying the physical three-dimensional space is introduces, and an original geometric treatment (admitting visualization) of some formerly considered abstract functions (mechanical action, space-time interval) are suggested.
Warchalowski, Wiktor; Krawczyk, Malgorzata J.
2017-03-01
We found the Lindenmayer systems for line graphs built on selected fractals. We show that the fractal dimension of such obtained graphs in all analysed cases is the same as for their original graphs. Both for the original graphs and for their line graphs we identified classes of nodes which reflect symmetry of the graph.
Perepelitsa, VA; Sergienko, [No Value; Kochkarov, AM
1999-01-01
Definitions of prefractal and fractal graphs are introduced, and they are used to formulate mathematical models in different fields of knowledge. The topicality of fractal-graph recognition from the point of view, of fundamental improvement in the efficiency of the solution of algorithmic problems i
Fractal and Chaos Characteristics in Rock Milled Process
Directory of Open Access Journals (Sweden)
Chenxu LUO
2013-07-01
Full Text Available In order to research the mechanism and to reveal the natural characteristics in rock milled process, the milling load was deemed to be a component of the deterministic nonlinear dissipation system. The characteristic factor model of milling load was built on the basis of time delay method, and the phase-space of milling load was rebuilt. the dimensional phase-type of broken attractor was described. The results indicated that the broken attractor was a fractal set ,which acquired through the scale conversion of phase-space developed by each dimension. The relevant dimension of broken attractor can be as the identification to reflect the change of rock broken mechanism. And the Lyapunov exponential spectrum and the maximum Lyapunov exponent were acquired by confirming the system reconstruction dimension. The chaos phenomenon was existed in the rock milled process, which provides the basis for building the deterministic model of rock milled process.
THREE-DIMENSIONAL FLOW AROUND TWO TANDEM CIRCULAR CYLINDERS WITH VARIOUS SPACING AT Re=220
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
The flow around two tandem circular cylinders was studied by a three-dimensional numerical simulation of the Navier-Stokes equations at Re=220. The improved virtual boundary method was applied to model the no-slip boundary condition of the cylinders. The results show that as the spacing ratio L/D ≥4, the three dimensionality occurs in the wake. When L/D≤3.5 the wake keeps a two-dimensional state at the Reynolds number Re=220. The critical spacing for the appearance of three-dimensional instability obtained is at the range 3.5＜L/D＜4, similar to the critical spacing found in two-dimensional case. Two sources of instability from upstream and downstream cylinder generate a complicated vortex structures in the wake, investigated by streamlines topology analysis in the streamwise plane. Many other interesting problems were also addressed in this paper.
Fractal images induce fractal pupil dilations and constrictions.
Moon, P; Muday, J; Raynor, S; Schirillo, J; Boydston, C; Fairbanks, M S; Taylor, R P
2014-09-01
Fractals are self-similar structures or patterns that repeat at increasingly fine magnifications. Research has revealed fractal patterns in many natural and physiological processes. This article investigates pupillary size over time to determine if their oscillations demonstrate a fractal pattern. We predict that pupil size over time will fluctuate in a fractal manner and this may be due to either the fractal neuronal structure or fractal properties of the image viewed. We present evidence that low complexity fractal patterns underlie pupillary oscillations as subjects view spatial fractal patterns. We also present evidence implicating the autonomic nervous system's importance in these patterns. Using the variational method of the box-counting procedure we demonstrate that low complexity fractal patterns are found in changes within pupil size over time in millimeters (mm) and our data suggest that these pupillary oscillation patterns do not depend on the fractal properties of the image viewed.
Smoothness spaces of higher order on lower dimensional subsets of the Euclidean space
Ihnatsyeva, Lizaveta
2011-01-01
We study Sobolev type spaces defined in terms of sharp maximal functions on Ahlfors regular subsets of the Euclidean space and the relation between these spaces and traces of classical Sobolev spaces.
Smitha, K A; Gupta, A K; Jayasree, R S
2015-09-07
Glioma, the heterogeneous tumors originating from glial cells, generally exhibit varied grades and are difficult to differentiate using conventional MR imaging techniques. When this differentiation is crucial in the disease prognosis and treatment, even the advanced MR imaging techniques fail to provide a higher discriminative power for the differentiation of malignant tumor from benign ones. A powerful image processing technique applied to the imaging techniques is expected to provide a better differentiation. The present study focuses on the fractal analysis of fluid attenuation inversion recovery MR images, for the differentiation of glioma. For this, we have considered the most important parameters of fractal analysis, fractal dimension and lacunarity. While fractal analysis assesses the malignancy and complexity of a fractal object, lacunarity gives an indication on the empty space and the degree of inhomogeneity in the fractal objects. Box counting method with the preprocessing steps namely binarization, dilation and outlining was used to obtain the fractal dimension and lacunarity in glioma. Statistical analysis such as one-way analysis of variance and receiver operating characteristic (ROC) curve analysis helped to compare the mean and to find discriminative sensitivity of the results. It was found that the lacunarity of low and high grade gliomas vary significantly. ROC curve analysis between low and high grade glioma for fractal dimension and lacunarity yielded 70.3% sensitivity and 66.7% specificity and 70.3% sensitivity and 88.9% specificity, respectively. The study observes that fractal dimension and lacunarity increases with an increase in the grade of glioma and lacunarity is helpful in identifying most malignant grades.
Acoustic space dimensionality selection and combination using the maximum entropy principle
Abdel-Haleem, Yasser H.; Renals, Steve; Lawrence, Neil D.
2004-01-01
In this paper we propose a discriminative approach to acoustic space dimensionality selection based on maximum entropy modelling. We form a set of constraints by composing the acoustic space with the space of phone classes, and use a continuous feature formulation of maximum entropy modelling to select an optimal feature set. The suggested approach has two steps: (1) the selection of the best acoustic space that efficiently and economically represents the acoustic data and its variability;...
On the finite-dimensional PUA representations of the Shubnikov space groups
Broek, van den P.M.
1977-01-01
The finite-dimensional PUA epresentations of the Shubnikov space groups are discussed using the method of generalised induction given by Shaw and Lever. In particular we derive expressions for the calculation of the little groups.
Food-web structure in low- and high-dimensional trophic niche spaces
Rossberg, Axel G.; Brännström, Åke; Dieckmann, Ulf
2010-01-01
A question central to modelling and, ultimately, managing food webs concerns the dimensionality of trophic niche space, that is, the number of independent traits relevant for determining consumer–resource links. Food-web topologies can often be interpreted by assuming resource traits to be specified by points along a line and each consumer's diet to be given by resources contained in an interval on this line. This phenomenon, called intervality, has been known for 30 years and is widely acknowledged to indicate that trophic niche space is close to one-dimensional. We show that the degrees of intervality observed in nature can be reproduced in arbitrary-dimensional trophic niche spaces, provided that the processes of evolutionary diversification and adaptation are taken into account. Contrary to expectations, intervality is least pronounced at intermediate dimensions and steadily improves towards lower- and higher-dimensional trophic niche spaces. PMID:20462875
Indian Academy of Sciences (India)
K D Patil; S H Ghate; R V Saraykar
2001-04-01
We consider a collapsing spherically symmetric inhomogeneous dust cloud in higher dimensional space-time. We show that the central singularity of collapse can be a strong curvature or a weak curvature naked singularity depending on the initial density distribution.
Construction of Fractal Surfaces by Recurrent Fractal Interpolation Curves
Yun, Chol-Hui; O., Hyong-chol; Choi, Hui-chol
2013-01-01
A method to construct fractal surfaces by recurrent fractal curves is provided. First we construct fractal interpolation curves using a recurrent iterated functions system(RIFS) with function scaling factors and estimate their box-counting dimension. Then we present a method of construction of wider class of fractal surfaces by fractal curves and Lipschitz functions and calculate the box-counting dimension of the constructed surfaces. Finally, we combine both methods to have more flexible con...
Fractal-Based Exponential Distribution of Urban Density and Self-Affine Fractal Forms of Cities
Chen, Yanguang
2016-01-01
Urban population density always follows the exponential distribution and can be described with Clark's model. Because of this, the spatial distribution of urban population used to be regarded as non-fractal pattern. However, Clark's model differs from the exponential function in mathematics because that urban population is distributed on the fractal support of landform and land-use form. By using mathematical transform and empirical evidence, we argue that there are self-affine scaling relations and local power laws behind the exponential distribution of urban density. The scale parameter of Clark's model indicating the characteristic radius of cities is not a real constant, but depends on the urban field we defined. So the exponential model suggests local fractal structure with two kinds of fractal parameters. The parameters can be used to characterize urban space filling, spatial correlation, self-affine properties, and self-organized evolution. The case study of the city of Hangzhou, China, is employed to ...
Hörmander multipliers on two-dimensional dyadic Hardy spaces
Daly, J.; Fridli, S.
2008-12-01
In this paper we are interested in conditions on the coefficients of a two-dimensional Walsh multiplier operator that imply the operator is bounded on certain of the Hardy type spaces Hp, 0Dokl. Akad. Nauk SSSR 109 (1956) 701-703; S.G. Mihlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, 1965]. In this paper we extend these results to the two-dimensional dyadic Hardy spaces.
Quantum scattering theory of a single-photon Fock state in three-dimensional spaces.
Liu, Jingfeng; Zhou, Ming; Yu, Zongfu
2016-09-15
A quantum scattering theory is developed for Fock states scattered by two-level systems in three-dimensional free space. It is built upon the one-dimensional scattering theory developed in waveguide quantum electrodynamics. The theory fully quantizes the incident light as Fock states and uses a non-perturbative method to calculate the scattering matrix.
Vibration modes of 3n-gaskets and other fractals
Energy Technology Data Exchange (ETDEWEB)
Bajorin, N; Chen, T; Dagan, A; Emmons, C; Hussein, M; Khalil, M; Mody, P; Steinhurst, B; Teplyaev, A [Department of Mathematics, University of Connecticut, Storrs CT 06269 (United States)
2008-01-11
We rigorously study eigenvalues and eigenfunctions (vibration modes) on the class of self-similar symmetric finitely ramified fractals, which include the Sierpinski gasket and other 3n-gaskets. We consider the classical Laplacian on fractals which generalizes the usual one-dimensional second derivative, is the generator of the self-similar diffusion process, and has possible applications as the quantum Hamiltonian. We develop a theoretical matrix analysis, including analysis of singularities, which allows us to compute eigenvalues, eigenfunctions and their multiplicities exactly. We support our theoretical analysis by symbolic and numerical computations. Our analysis, in particular, allows the computation of the spectral zeta function on fractals and the limiting distribution of eigenvalues (i.e., integrated density of states). We consider such examples as the level-3 Sierpinski gasket, a fractal 3-tree, and the diamond fractal.
Menger sponge-like fractal body created by a novel template method.
Mayama, H; Tsujii, K
2006-09-28
We have established experimental strategies on how to create a Menger sponge-like fractal body and how to control its fractal dimension. The essence was to utilize alkylketene dimer (AKD), which spontaneously forms super-water-repellent fractal surface. We prepared "fractal AKD particles" with fractal surface structure as templates of pores in fractal body. The fractal body was synthesized by filling the remained space between the packed template particles with a tetramethyl orthosilicate solution, solidifying it by the sol-gel process, and removing the template by calcinations. We have succeeded in systematically creating fractal bodies of silica with different cross-sectional fractal dimensions D(cs)=1.87, 1.84, and 1.80 using "fractal template particles" compressed under the ratio=1.0, 2.0, and 3.0, respectively. We also discussed the possibilities of their fractal geometries in comparison with mathematical models. We concluded that the created fractal bodies were close to a Menger sponge and its modified one. Our experimental strategy allows us to design fractality of porous materials.
Geometry dependence of 2-dimensional space-charge-limited currents
De Visschere, Patrick
2016-01-01
The space-charge-limited current in a zero thickness planar thin film depends on the geometry of the electrodes. We present a theory which is to a large extent analytical and applicable to many different lay-outs. We show that a space-charge-limited current can only be sustained if the emitting electrode induces a singularity in the field and if the singularity induced by the collecting electrode is not too strong. For those lay-outs where no space-charge-limited current can be sustained for a zero thickness film, the real thickness of the film must be taken into account using a numerical model.
Lee, Jenny Hyunjung; McDonnell, Kevin T; Zelenyuk, Alla; Imre, Dan; Mueller, Klaus
2013-07-11
Although the Euclidean distance does well in measuring data distances within high-dimensional clusters, it does poorly when it comes to gauging inter-cluster distances. This significantly impacts the quality of global, low-dimensional space embedding procedures such as the popular multi-dimensional scaling (MDS) where one can often observe non-intuitive layouts. We were inspired by the perceptual processes evoked in the method of parallel coordinates which enables users to visually aggregate the data by the patterns the polylines exhibit across the dimension axes. We call the path of such a polyline its structure and suggest a metric that captures this structure directly in high-dimensional space. This allows us to better gauge the distances of spatially distant data constellations and so achieve data aggregations in MDS plots that are more cognizant of existing high-dimensional structure similarities. Our bi-scale framework distinguishes far-distances from near-distances. The coarser scale uses the structural similarity metric to separate data aggregates obtained by prior classification or clustering, while the finer scale employs the appropriate Euclidean distance.
Attractive and repulsive quantum forces from dimensionality of space
DEFF Research Database (Denmark)
Bialynicki-Birula, I.; Cirone, M.A.; Dahl, Jens Peder
2002-01-01
Two particles of identical mass attract and repel each other even when there exist no classical external forces and their average relative momentum vanishes. This quantum force depends crucially on the number of dimensions of space.......Two particles of identical mass attract and repel each other even when there exist no classical external forces and their average relative momentum vanishes. This quantum force depends crucially on the number of dimensions of space....
Objetos fractales y arquitectura
MARTÍNEZ REQUENA, CELIA ANA
2015-01-01
Este trabajo final de grado versa acerca de la fractalidad y su posible aplicación arquitectónica. Se parte del concepto de fractal quedándose con la idea de que “un fractal es un diseño que se repite indefinidamente hacia el infinito cada vez a escala menor” y se presentan los diferentes conjuntos haciendo especial hincapié en los fractales clásicos. La fractalidad se puede apreciar en la naturaleza (p.e: un árbol tiene un tronco, este se divide en ramas, cada una de ellas en ...
Objetos fractales y arquitectura
MARTÍNEZ REQUENA, CELIA ANA
2015-01-01
Este trabajo final de grado versa acerca de la fractalidad y su posible aplicación arquitectónica. Se parte del concepto de fractal quedándose con la idea de que “un fractal es un diseño que se repite indefinidamente hacia el infinito cada vez a escala menor” y se presentan los diferentes conjuntos haciendo especial hincapié en los fractales clásicos. La fractalidad se puede apreciar en la naturaleza (p.e: un árbol tiene un tronco, este se divide en ramas, cada una de ellas en ...
Data analysis in high-dimensional sparse spaces
DEFF Research Database (Denmark)
Clemmensen, Line Katrine Harder
The present thesis considers data analysis of problems with many features in relation to the number of observations (large p, small n problems). The theoretical considerations for such problems are outlined including the curses and blessings of dimensionality, and the importance of dimension...... reduction. In this context the trade off between a rich solution which answers the questions at hand and a simple solution which generalizes to unseen data is described. For all of the given data examples labelled output exists and the analyses are therefore limited to supervised settings. Three novel...... classification techniques for high-dimensional problems are presented: Sparse discriminant analysis, sparse mixture discriminant analysis and orthogonality constrained support vector machines. The first two introduces sparseness to the well known linear and mixture discriminant analysis and thereby provide low...
APPLICATIONS OF FRACTIONAL EXTERIOR DIFFERENTIAL IN THREE-DIMENSIONAL SPACE
Institute of Scientific and Technical Information of China (English)
陈勇; 闫振亚; 张鸿庆
2003-01-01
A brief survey of fractional calculus and fractional differential forms was firstly given. The fractional exterior transition to curvilinear coordinate at the origin were discussed and the two coordinate transformations for the fractional differentials for three-dimensional Cartesian coordinates to spherical and cylindrical coordinates are obtained, respectively. In particular, for v = m = 1 , the usual exterior transformations, between the spherical coordinate and Cartesian coordinate, as well as the cylindrical coordinate and Cartesian coordinate, are found respectively, from fractional exterior transformation.
Chaotic behaviour of nonlinear coupled reaction–diffusion system in four-dimensional space
Indian Academy of Sciences (India)
Li Zhang; Shutang Liu; Chenglong Yu
2014-06-01
In recent years, nonlinear coupled reaction–diffusion (CRD) system has been widely investigated by coupled map lattice method. Previously, nonlinear behaviour was observed dynamically when one or two of the three variables in the discrete system change. In this paper, we consider the chaotic behaviour when three variables change, which is called as four-dimensional chaos. When two parameters in the discrete system are unknown, we first give the existing condition of the chaos in four-dimensional space by the generalized definitions of spatial periodic orbits and spatial chaos. In addition, the chaotic behaviour will vary with the parameters. Then we propose a generalized Lyapunov exponent in four-dimensional space to characterize the different effects of parameters on the chaotic behaviour, which has not been studied in detail. In order to verify the chaotic behaviour of the system and the different effects clearly, we simulate the dynamical behaviour in two- and three-dimensional spaces.
The Schwartz Space: Tools for Quantum Mechanics and Infinite Dimensional Analysis
Directory of Open Access Journals (Sweden)
Jeremy Becnel
2015-06-01
Full Text Available An account of the Schwartz space of rapidly decreasing functions as a topological vector space with additional special structures is presented in a manner that provides all the essential background ideas for some areas of quantum mechanics along with infinite-dimensional distribution theory.
Approximate controllability of infinite dimensional linear systems in nonreflexive state spaces
Institute of Scientific and Technical Information of China (English)
Xin YU; Chao XU
2005-01-01
This paper deals with the problem of approximate controllability of infinite dimensional linear systems in nonreflexive state spaces.A necessary and sufficient condition for approximate controllability via Lp([0,T],U),1≤p<∞ is obtained,where Lp([0,T],U) is the control function space.
Thermal field diffusion in one, two and three-dimensional half space
Directory of Open Access Journals (Sweden)
Cividjian Grigore A.
2011-01-01
Full Text Available The diffusion of suddenly occurring local high temperature in homogeneous half-infinite space is studied in the cases of one, two and three-dimensional half space. Comparison of the three cases is made. Applications of theoretically analyzed models are suggested. Errors induced by assumptions are evaluated.
CFSBC:Clustering in High-Dimensional Space Based on Closed Frequent Item Set
Institute of Scientific and Technical Information of China (English)
NI Wei-wei; SUN Zhi-hui
2004-01-01
Clustering in high-dimensional space is an important domain in data mining.It is the process of discovering groups in a high-dimensional dataset, in such way, that the similarity between the elements of the same cluster is maximum and between different clusters is minimal.Many clustering algorithms are not applicable to high-dimensional space for its sparseness and decline properties.Dimensionality reduction is an effective method to solve this problem.The paper proposes a novel clustering algorithm CFSBC based on closed frequent itemsets derived from association rule mining, which can get the clustering attributes with high efficiency.The algorithm has several advantages.First, it deals effectively with the problem of dimensionality reduction.Second, it is applicable to different kinds of attributes.Third, it is suitable for very large data sets.Experiment shows that the proposed algorithm is effective and efficient.
Interstellar extinction by fractal polycrystalline graphite clusters?
Andersen, A C; Pustovit, V N; Niklasson, G A
2001-01-01
Certain dust particles in space are expected to appear as clusters of individual grains. The morphology of these clusters could be fractal or compact. To determine how these structural features would affect the interpretation of the observed interstellar extinction peak at $\\sim 4.6~\\mu$m, we have calculated the extinction by compact and fractal polycrystalline graphite clusters consisting of touching identical spheres. We compare three general methods for computing the extinction of the clusters, namely, a rigorous solution and two different discrete-dipole approximation methods.
Three dimensional monocular human motion analysis in end-effector space
DEFF Research Database (Denmark)
Hauberg, Søren; Lapuyade, Jerome; Engell-Nørregård, Morten Pol
2009-01-01
In this paper, we present a novel approach to three dimensional human motion estimation from monocular video data. We employ a particle filter to perform the motion estimation. The novelty of the method lies in the choice of state space for the particle filter. Using a non-linear inverse kinemati...... solver allows us to perform the filtering in end-effector space. This effectively reduces the dimensionality of the state space while still allowing for the estimation of a large set of motions. Preliminary experiments with the strategy show good results compared to a full-pose tracker....
Transmission and reflection properties of terahertz fractal metamaterials
DEFF Research Database (Denmark)
Malureanu, Radu; Lavrinenko, Andrei; Cooke, David
2010-01-01
We use THz time-domain spectroscopy to investigate transmission and reflection properties of metallic fractal metamaterial structures. We observe loss of free-space energy at certain resonance frequencies, indicating excitation of surface modes of the metamaterial.......We use THz time-domain spectroscopy to investigate transmission and reflection properties of metallic fractal metamaterial structures. We observe loss of free-space energy at certain resonance frequencies, indicating excitation of surface modes of the metamaterial....
Critical behavior of the Gaussian model on fractal lattices in external magnetic field
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
For inhomogeneous lattices we generalize the classical Gaussian model, i.e. it is proposed that the Gaussian type distribution constant and the external magnetic field of site i in this model depend on the coordination number qi of site i, and that the relation bqi/bqj=qi/qj holds among bqi's, where bqi is the Gaussian type distribution constant of site i. Using the decimation real-space renormalization group following the spin-rescaling method, the critical points and critical exponents of the Gaussian model are calculated on some Koch type curves and a family of the diamond-type hierarchical (or DH) lattices. At the critical points, it is found that the nearest-neighbor interaction and the magnetic field of site i can be expressed in the form K*=bqi/qi and h*qi=0, respectively. It is also found that most critical exponents depend on the fractal dimensionality of a fractal system. For the family of the DH lattices, the results are identical with the exact results on translation symmetric lattices, and if the fractal dimensionality df=4, the Gaussian model and the mean field theories give the same results.
Salzman, Oren; Halperin, Dan
2012-01-01
We extend our study of Motion Planning via Manifold Samples (MMS), a general algorithmic framework that combines geometric methods for the exact and complete analysis of low-dimensional configuration spaces with sampling-based approaches that are appropriate for higher dimensions. The framework explores the configuration space by taking samples that are entire low-dimensional manifolds of the configuration space capturing its connectivity much better than isolated point samples. The contributions of this paper are as follows: (i) We present a recursive application of MMS in a six-dimensional configuration space, enabling the coordination of two polygonal robots translating and rotating amidst polygonal obstacles. In the adduced experiments for the more demanding test cases MMS clearly outperforms PRM, with over 20-fold speedup in a coordination-tight setting. (ii) A probabilistic completeness proof for the most prevalent case, namely MMS with samples that are affine subspaces. (iii) A closer examination of th...
Relativistic Fractal Cosmologies
Ribeiro, Marcelo B
2009-01-01
This article reviews an approach for constructing a simple relativistic fractal cosmology whose main aim is to model the observed inhomogeneities of the distribution of galaxies by means of the Lemaitre-Tolman solution of Einstein's field equations for spherically symmetric dust in comoving coordinates. This model is based on earlier works developed by L. Pietronero and J.R. Wertz on Newtonian cosmology, whose main points are discussed. Observational relations in this spacetime are presented, together with a strategy for finding numerical solutions which approximate an averaged and smoothed out single fractal structure in the past light cone. Such fractal solutions are shown, with one of them being in agreement with some basic observational constraints, including the decay of the average density with the distance as a power law (the de Vaucouleurs' density power law) and the fractal dimension in the range 1 <= D <= 2. The spatially homogeneous Friedmann model is discussed as a special case of the Lemait...
Directory of Open Access Journals (Sweden)
Alexander J. Bies
2016-07-01
Full Text Available Two measures are commonly used to describe scale-invariant complexity in images: fractal dimension (D and power spectrum decay rate (β. Although a relationship between these measures has been derived mathematically, empirical validation across measurements is lacking. Here, we determine the relationship between D and β for 1- and 2-dimensional fractals. We find that for 1-dimensional fractals, measurements of D and β obey the derived relationship. Similarly, in 2-dimensional fractals, measurements along any straight-line path across the fractal’s surface obey the mathematically derived relationship. However, the standard approach of vision researchers is to measure β of the surface after 2-dimensional Fourier decomposition rather than along a straight-line path. This surface technique provides measurements of β that do not obey the mathematically derived relationship with D. Instead, this method produces values of β that imply that the fractal’s surface is much smoother than the measurements along the straight lines indicate. To facilitate communication across disciplines, we provide empirically derived equations for relating each measure of β to D. Finally, we discuss implications for future research on topics including stress reduction and the perception of motion in the context of a generalized equation relating β to D.
Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces
Jacob, Birgit
2012-01-01
This book provides a self-contained introduction to the theory of infinite-dimensional systems theory and its applications to port-Hamiltonian systems. The textbook starts with elementary known results, then progresses smoothly to advanced topics in current research. Many physical systems can be formulated using a Hamiltonian framework, leading to models described by ordinary or partial differential equations. For the purpose of control and for the interconnection of two or more Hamiltonian systems it is essential to take into account this interaction with the environment. This book is the fir
Fractal growth in impurity-controlled solidification in lipid monolayers
DEFF Research Database (Denmark)
Fogedby, Hans C.; Sørensen, Erik Schwartz; Mouritsen, Ole G.
1987-01-01
A simple two-dimensional microscopic model is proposed to describe solidifcation processes in systems with impurities which are miscible only in the fluid phase. Computer simulation of the model shows that the resulting solids are fractal over a wide range of impurity concentrations and impurity...... diffusional constants. A fractal-forming mechanism is suggested for impurity-controlled solidification which is consistent with recent experimental observations of fractal growth of solid phospholipid domains in monolayers. The Journal of Chemical Physics is copyrighted by The American Institute of Physics....
Fractal design concepts for stretchable electronics.
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
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.
Vibrations of strongly irregular or fractal resonators
Sapoval, B.; Gobron, Th.
1993-05-01
It is shown on a specific example that fractal boundary conditions drastically alter the properties of wave excitations in space. The low-frequency part of the vibration spectrum of a finite-range fractal drum is computed using an analogy between the Helmoltz equation and the diffusion equation. The irregularity of the frontier is found to influence strongly the density of states at low frequency. The fractal perimeter generates a specific screening effect. Very near the frontier, the decrease of the wave form is related directly to the behavior of the harmonic measure. The possibility of localization of the vibrations is qualitatively discussed and we show that localized modes may exist at low frequencies if the geometrical structures possess narrow paths. Possible application of these results to the interpretation of thermal properties of binary glasses is briefly discussed.
Edge effect causes apparent fractal correlation dimension of uniform spatial raindrop distribution
Directory of Open Access Journals (Sweden)
R. Uijlenhoet
2009-04-01
Full Text Available Lovejoy and Schertzer (1990a presented a statistical analysis of blotting paper observations of the (two-dimensional spatial distribution of raindrop stains. They found empirical evidence for the fractal scaling behavior of raindrops in space, with potentially far-reaching implications for rainfall microphysics and radar meteorology. In particular, the fractal correlation dimensions determined from their blotting paper observations led them to conclude that "drops are (hierarchically clustered" and that "inhomogeneity in rain is likely to extend down to millimeter scales". Confirming previously reported Monte Carlo simulations, we demonstrate analytically that the claims based on this analysis need to be reconsidered, as fractal correlation dimensions similar to the ones reported (i.e. smaller than the value of two expected for uniformly distributed raindrops can result from instrumental artifacts (edge effects in otherwise homogeneous Poissonian rainfall. Hence, the results of the blotting paper experiment are not statistically significant enough to reject the Poisson homogeneity hypothesis in favor of a fractal description of the discrete nature of rainfall. Our analysis is based on an analytical expression for the expected overlap area between a circle and a square, when the circle center is randomly (uniformly distributed inside the square. The derived expression (πr^{2}−8r^{3}/3+r^{4}/2, where r denotes the ratio between the circle radius and the side of the square can be used as a reference curve against which to test the statistical significance of fractal correlation dimensions determined from spatial point patterns, such as those of raindrops and rainfall cells.
Trabajando fractales con Winlogo
Sabogal, Sonia; Arenas, Gilberto
2007-01-01
Después de una breve introducción en la cual se establecerán algunos conceptos teóricos básicos de la geometría fractal, se realizarán talleres en los cuales, con ayuda de las herramientas que trabaja el software WinLogo, se construirán diversos fractales, analizando sus principales características (autosimilitud, dimensión, etc.)
Turcotte, Donald L.
Tectonic processes build landforms that are subsequently destroyed by erosional processes. Landforms exhibit fractal statistics in a variety of ways; examples include (1) lengths of coast lines; (2) number-size statistics of lakes and islands; (3) spectral behavior of topography and bathymetry both globally and locally; and (4) branching statistics of drainage networks. Erosional processes are dominant in the development of many landforms on this planet, but similar fractal statistics are also applicable to the surface of Venus where minimal erosion has occurred. A number of dynamical systems models for landforms have been proposed, including (1) cellular automata; (2) diffusion limited aggregation; (3) self-avoiding percolation; and (4) advective-diffusion equations. The fractal statistics and validity of these models will be discussed. Earthquakes also exhibit fractal statistics. The frequency-magnitude statistics of earthquakes satisfy the fractal Gutenberg-Richter relation both globally and locally. Earthquakes are believed to be a classic example of self-organized criticality. One model for earthquakes utilizes interacting slider-blocks. These slider block models have been shown to behave chaotically and to exhibit self-organized criticality. The applicability of these models will be discussed and alternative approaches will be presented. Fragmentation has been demonstrated to produce fractal statistics in many cases. Comminution is one model for fragmentation that yields fractal statistics. It has been proposed that comminution is also responsible for much of the deformation in the earth's crust. The brittle disruption of the crust and the resulting earthquakes present an integrated problem with many fractal aspects.
Influence Factors of Fractal Characterization of Reciprocating Sliding Wear Surfaces
Institute of Scientific and Technical Information of China (English)
周新聪; 冯伟; 严新平; 萧汉梁
2004-01-01
The principal purpose of this paper is to investigate influence factors of fractal characterization of reciprocating sliding wear surfaces.The wear testing was completed to simulate the real running condition of the diesel engine 8NVD48A-2U.The test results of wear surface morphology dimension characterization show that wear surface profiles have statistical self-affine fractal characteristics.In general, there are no effects of the profilometer sampling spacing and sampling length and evaluation length on the fractal dimensions of the surfaces.However, if the evaluation length is too short, the structure function logarithm of the surface profile is scattered.The sampling length acting as a filter is an important part of the fractal dimension measurement.If the sampling length is too short, the evaluation of the fractal dimension will have a larger standard deviation.The continuous wavelet transform can be used to improve surface profile dimension characterization.
[Recent progress of research and applications of fractal and its theories in medicine].
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.
Institute of Scientific and Technical Information of China (English)
Yi Zhuang; Yue-Ting Zhuang; Fei Wu
2007-01-01
Due to the famous dimensionality curse problem, search in a high-dimensional space is considered as a "hard" problem. In this paper, a novel composite distance transformation method, which is called CDT, is proposed to support a fast k-nearest-neighbor (k-NN) search in high-dimensional spaces. In CDT, all (n) data points are first grouped into some clusters by a k-Means clustering algorithm. Then a composite distance key of each data point is computed. Finally, these index keys of such n data points are inserted by a partition-based B+-tree. Thus, given a query point, its k-NN search in high-dimensional spaces is transformed into the search in the single dimensional space with the aid of CDT index. Extensive performance studies are conducted to evaluate the effectiveness and efficiency of the proposed scheme. Our results showthat this method outperforms the state-of-the-art high-dimensional search techniques, such as the X-Tree, VA-file, iDistance and NB-Tree.
On Space Efficient Two Dimensional Range Minimum Data Structures
DEFF Research Database (Denmark)
Davoodi, Pooya; Brodal, Gerth Stølting; Rao, S. Srinivasa
2010-01-01
, the lower bound is tight up to a constant factor. In two dimensions, we complement the lower bound with an indexing data structure of size O(N/c) bits additional space which can be preprocessed in O(N) time and achieves O(clog2 c) query time. For c = O(1), this is the first O(1) query time algorithm using...... optimal O(N) bits additional space. For the case where queries can not probe A, we give a data structure of size O(N· min {m,logn}) bits with O(1) query time, assuming m ≤ n. This leaves a gap to the lower bound of Ω(Nlogm) bits for this version of the problem....
On High Dimensional Searching Spaces and Learning Methods
DEFF Research Database (Denmark)
Yazdani, Hossein; Ortiz-Arroyo, Daniel; Choros, Kazimierz
2017-01-01
In data science, there are important parameters that affect the accuracy of the algorithms used. Some of these parameters are: the type of data objects, the membership assignments, and distance or similarity functions. In this chapter we describe different data types , membership functions...... , and similarity functions and discuss the pros and cons of using each of them. Conventional similarity functions evaluate objects in the vector space. Contrarily, Weighted Feature Distance (WFD) functions compare data objects in both feature and vector spaces, preventing the system from being affected by some...... dominant features. Traditional membership functions assign membership values to data objects but impose some restrictions. Bounded Fuzzy Possibilistic Method (BFPM) makes possible for data objects to participate fully or partially in several clusters or even in all clusters. BFPM introduces intervals...
Topological entropy and renormalization group flow in 3-dimensional spherical spaces
Energy Technology Data Exchange (ETDEWEB)
Asorey, M. [Departamento de Física Teórica, Universidad de Zaragoza,E-50009 Zaragoza (Spain); Beneventano, C.G. [Departamento de Física, Universidad Nacional de La Plata,Instituto de Física de La Plata, CONICET-Universidad Nacional de La Plata,C.C. 67, 1900 La Plata (Argentina); Cavero-Peláez, I. [Departamento de Física Teórica, Universidad de Zaragoza,E-50009 Zaragoza (Spain); CUD,E-50090, Zaragoza (Spain); D’Ascanio, D.; Santangelo, E.M. [Departamento de Física, Universidad Nacional de La Plata,Instituto de Física de La Plata, CONICET-Universidad Nacional de La Plata,C.C. 67, 1900 La Plata (Argentina)
2015-01-15
We analyze the renormalization group (RG) flow of the temperature independent term of the entropy in the high temperature limit β/a≪1 of a massive field theory in 3-dimensional spherical spaces, M{sub 3}, with constant curvature 6/a{sup 2}. For masses lower than ((2π)/β), this term can be identified with the free energy of the same theory on M{sub 3} considered as a 3-dimensional Euclidean space-time. The non-extensive part of this free energy, S{sub hol}, is generated by the holonomy of the spatial metric connection. We show that for homogeneous spherical spaces the holonomy entropy S{sub hol} decreases monotonically when the RG scale flows to the infrared. At the conformal fixed points the values of the holonomy entropy do coincide with the genuine topological entropies recently introduced. The monotonic behavior of the RG flow leads to an inequality between the topological entropies of the conformal field theories connected by such flow, i.e. S{sub top}{sup UV}>S{sub top}{sup IR}. From a 3-dimensional viewpoint the same term arises in the 3-dimensional Euclidean effective action and has the same monotonic behavior under the RG group flow. We conjecture that such monotonic behavior is generic, which would give rise to a 3-dimensional generalization of the c-theorem, along the lines of the 2-dimensional c-theorem and the 4-dimensional a-theorem. The conjecture is related to recent formulations of the F-theorem. In particular, the holonomy entropy on lens spaces is directly related to the topological Rényi entanglement entropy on disks of 2-dimensional flat spaces.
Topological entropy and renormalization group flow in 3-dimensional spherical spaces
Asorey, M.; Beneventano, C. G.; Cavero-Peláez, I.; D'Ascanio, D.; Santangelo, E. M.
2015-01-01
We analyze the renormalization group (RG) flow of the temperature independent term of the entropy in the high temperature limit β/a ≪ 1 of a massive field theory in 3-dimensional spherical spaces, M 3, with constant curvature 6 /a 2. For masses lower than , this term can be identified with the free energy of the same theory on M 3 considered as a 3-dimensional Euclidean space-time. The non-extensive part of this free energy, S hol, is generated by the holonomy of the spatial metric connection. We show that for homogeneous spherical spaces the holonomy entropy S hol decreases monotonically when the RG scale flows to the infrared. At the conformal fixed points the values of the holonomy entropy do coincide with the genuine topological entropies recently introduced. The monotonic behavior of the RG flow leads to an inequality between the topological entropies of the conformal field theories connected by such flow, i.e. S {top/ UV } > S {top/ IR }. From a 3-dimensional viewpoint the same term arises in the 3-dimensional Euclidean effective action and has the same monotonic behavior under the RG group flow. We conjecture that such monotonic behavior is generic, which would give rise to a 3-dimensional generalization of the c-theorem, along the lines of the 2-dimensional c-theorem and the 4-dimensional a-theorem. The conjecture is related to recent formulations of the F -theorem. In particular, the holonomy entropy on lens spaces is directly related to the topological Rényi entanglement entropy on disks of 2-dimensional flat spaces.
A construction of full qed using finite dimensional Hilbert space
Francis, Charles
2006-01-01
While causal perturbation theory and lattice regularisation allow treatment of the ultraviolet divergences in qed, they do not resolve the issues of constructive field theory, or show the validity of qed except as a perturbation theory. I present a rigorous construction of quantum and classical electrodynamics from fundamental principles of quantum theory. Hilbert space of dimension N is justified from statements about measurements with finite range and resolution. Using linear combinations o...
The Space Complexity of 2-Dimensional Approximate Range Counting
DEFF Research Database (Denmark)
Wei, Zhewei; Yi, Ke
2013-01-01
with additive error εn. A well-known solution for this problem is the ε-approximation. Informally speaking, an ε-approximation of P is a subset A ⊆ P that allows us to estimate the number of points in P ∩ R by counting the number of points in A ∩ R. It is known that an ε-approximation of size exists for any P......We study the problem of 2-dimensional orthogonal range counting with additive error. Given a set P of n points drawn from an n × n grid and an error parameter ε, the goal is to build a data structure, such that for any orthogonal range R, the data structure can return the number of points in P ∩ R...
Dimensional characterization of anesthesia dynamic in reconstructed embedding space.
Gifani, P; Rabiee, H R; Hashemi, M; Ghanbari, M
2007-01-01
The depth of anesthesia quantification has been one of the most research interests in the field of EEG signal processing and nonlinear dynamical analysis has emerged as a novel method for the study of complex systems in the past few decades. In this investigation we use the concept of nonlinear time series analysis techniques to reconstruct the attractor of anesthesia from EEG signal which have been obtained from different hypnotic states during surgery to give a characterization of the dimensional complexity of EEG by Correlation Dimension estimation. The dimension of the anesthesia strange attractor can be thought of as a measure of the degrees of freedom or the ;complexity' of the dynamics at different hypnotic levels. The results imply that for awaked state the correlation dimension is high, On the other hand, for light, moderate and deep hypnotic states these values decrease respectively; which means for anesthetized situation we expect lower correlation dimension.
Institute of Scientific and Technical Information of China (English)
Ren Xin-Cheng; Guo Li-Xin
2008-01-01
A normalized two-dimensional band-limited Weierstrass fractal function is used for modelling the dielectric rough surface. An analytic solution of the scattered field is derived based on the Kirchhoff approximation. The variance of scat-tering intensity is presented to study the fractal characteristics through theoretical analysis and numerical calculations. The important conclusion is obtained that the diffracted envelope slopes of scattering pattern can be approximated as a slope of linear equation. This conclusion will be applicable for solving the inverse problem of reconstructing rough surface and remote sensing.
TRANSIENT CHAOS AND FRACTAL STRUCTURES IN PLANETARY FEEDING ZONES
Energy Technology Data Exchange (ETDEWEB)
Kovács, T. [Also at University of Applied Sciences, Nagy Lajos kir. útja 1-9, H-1148 Budapest, Hungary. (Hungary); Regály, Zs. [Konkoly Observatory of the Hungarian Academy of Sciences, P.O. Box 67, H-1525 Budapest (Hungary)
2015-01-01
The circular restricted three-body problem is investigated in the context of accretion and scattering processes. In our model, a large number of identical non-interacting mass-less planetesimals are considered in the planar case orbiting a star-planet system. This description allows us to investigate the gravitational scattering and possible capture of the particles by the forming planetary embryo in a dynamical systems approach. Although the problem serves a large variety of complex motions, the results can be easily interpreted because of the low dimensionality of the phase space. We show that initial conditions define isolated regions of the disk, where planetesimals accrete or escape, which have, in fact, a fractal structure. The fractal geometry of these ''basins'' implies that the dynamics is very complex. Based on the calculated escape rates and escape times, it is also demonstrated that the planetary accretion rate is exponential for short times and follows a power law for longer integration. A new numerical calculation of the maximum mass that a planet can reach (described by the expression of the isolation mass) is also derived.
Fractal Weyl law for quantum fractal eigenstates.
Shepelyansky, D L
2008-01-01
The properties of the resonant Gamow states are studied numerically in the semiclassical limit for the quantum Chirikov standard map with absorption. It is shown that the number of such states is described by the fractal Weyl law, and their Husimi distributions closely follow the strange repeller set formed by classical orbits nonescaping in future times. For large matrices the distribution of escape rates converges to a fixed shape profile characterized by a spectral gap related to the classical escape rate.
Fractal energy spectrum of a polariton gas in a Fibonacci quasiperiodic potential.
Tanese, D; Gurevich, E; Baboux, F; Jacqmin, T; Lemaître, A; Galopin, E; Sagnes, I; Amo, A; Bloch, J; Akkermans, E
2014-04-11
We report on the study of a polariton gas confined in a quasiperiodic one-dimensional cavity, described by a Fibonacci sequence. Imaging the polariton modes both in real and reciprocal space, we observe features characteristic of their fractal energy spectrum such as the opening of minigaps obeying the gap labeling theorem and log-periodic oscillations of the integrated density of states. These observations are accurately reproduced solving an effective 1D Schrödinger equation, illustrating the potential of cavity polaritons as a quantum simulator in complex topological geometries.
Fractal Energy Spectrum of a Polariton Gas in a Fibonacci Quasiperiodic Potential
Tanese, D.; Gurevich, E.; Baboux, F.; Jacqmin, T.; Lemaètre, A.; Galopin, E.; Sagnes, I.; Amo, A.; Bloch, J.; Akkermans, E.
2014-04-01
We report on the study of a polariton gas confined in a quasiperiodic one-dimensional cavity, described by a Fibonacci sequence. Imaging the polariton modes both in real and reciprocal space, we observe features characteristic of their fractal energy spectrum such as the opening of minigaps obeying the gap labeling theorem and log-periodic oscillations of the integrated density of states. These observations are accurately reproduced solving an effective 1D Schrödinger equation, illustrating the potential of cavity polaritons as a quantum simulator in complex topological geometries.
On the sampling of serial sectioning technique for three dimensional space-filling grain structures:
Guoquan Liu; Haibo Yu
2000-01-01
Serial sectioning technique provides plenty of quantitative geometric information of the microstructure analyzed, including those unavailable from stereology with one- and two-dimensional probes. This may be why it used to be and is being continuously served as one of the most common and invaluable methods to study the size and the size distribution, the topology and the distribution of topology parameters, and even the shape of three-dimensional space filling grains or cells. On the other ha...
Gravitational collapse in higher-dimensional charged-Vaidya space-time
Indian Academy of Sciences (India)
Kishor D Patil
2003-03-01
We analyze here the gravitational collapse of higher-dimensional charged-Vaidya space-time. We show that singularities arising in a charged null ﬂuid in higher dimension are always naked violating at least strong cosmic censorship hypothesis (CCH), though not necessarily weak CCH. We show that earlier conclusions on the occurrence of naked singularities in four-dimensional case can be extended essentially in the same manner in 5D case also.
Quantum algebras for maximal motion groups of n-dimensional flat spaces
Ballesteros, A; Del Olmo, M A; Santander, M
1994-01-01
An embedding method to get q-deformations for the non-semisimple algebras generating the motion groups of N-dimensional flat spaces is presented. This method gives a global and simultaneous scheme of q-deformation for all iso(p,q) algebras and for those ones obtained from them by some Inönü-Wigner contractions, such as the N--dimensional Euclidean, Poincaré and Galilei algebras.
Causal structures in the four dimensional Euclidean space
Pestov, I B; Pestov, Ivanhoe B.; Saha, Bijan
2005-01-01
It is shown that in the 4d Euclidean space there are two causal structures defined by the temporal field. One of them is well-known Minkowski spacetime. In this case the gravitational potential (the positive definite Riemann metric) and temporal field satisfy the Einstein equations with trivial energy-momentum tensor. However, in the case of the second causal structure the gravitational potential and temporal field should be connected with some nontrivial energy-momentum tensor. We consider the simplest case with energy-momentum tensor of the real scalar field and derive exact solution of the field equations. It can be viewed as the ground to consider the second causal structure on the equal footing with the Minkowski spacetime, i.e., as an object interesting from the physical point of view, especially in the framework of the field theory.
The fractal forest: fractal geometry and applications in forest science.
Nancy D. Lorimer; Robert G. Haight; Rolfe A. Leary
1994-01-01
Fractal geometry is a tool for describing and analyzing irregularity. Because most of what we measure in the forest is discontinuous, jagged, and fragmented, fractal geometry has potential for improving the precision of measurement and description. This study reviews the literature on fractal geometry and its applications to forest measurements.
Tournament screening cum EBIC for feature selection with high-dimensional feature spaces
Institute of Scientific and Technical Information of China (English)
2009-01-01
The feature selection characterized by relatively small sample size and extremely high-dimensional feature space is common in many areas of contemporary statistics.The high dimensionality of the feature space causes serious diffculties:(i) the sample correlations between features become high even if the features are stochastically independent;(ii) the computation becomes intractable.These diffculties make conventional approaches either inapplicable or ine?cient.The reduction of dimensionality of the feature space followed by low dimensional approaches appears the only feasible way to tackle the problem.Along this line,we develop in this article a tournament screening cum EBIC approach for feature selection with high dimensional feature space.The procedure of tournament screening mimics that of a tournament.It is shown theoretically that the tournament screening has the sure screening property,a necessary property which should be satisfied by any valid screening procedure.It is demonstrated by numerical studies that the tournament screening cum EBIC approach enjoys desirable properties such as having higher positive selection rate and lower false discovery rate than other approaches.
Functional connectivity among spikes in low dimensional space during working memory task in rat.
Directory of Open Access Journals (Sweden)
Mei Ouyang
Full Text Available Working memory (WM is critically important in cognitive tasks. The functional connectivity has been a powerful tool for understanding the mechanism underlying the information processing during WM tasks. The aim of this study is to investigate how to effectively characterize the dynamic variations of the functional connectivity in low dimensional space among the principal components (PCs which were extracted from the instantaneous firing rate series. Spikes were obtained from medial prefrontal cortex (mPFC of rats with implanted microelectrode array and then transformed into continuous series via instantaneous firing rate method. Granger causality method is proposed to study the functional connectivity. Then three scalar metrics were applied to identify the changes of the reduced dimensionality functional network during working memory tasks: functional connectivity (GC, global efficiency (E and casual density (CD. As a comparison, GC, E and CD were also calculated to describe the functional connectivity in the original space. The results showed that these network characteristics dynamically changed during the correct WM tasks. The measure values increased to maximum, and then decreased both in the original and in the reduced dimensionality. Besides, the feature values of the reduced dimensionality were significantly higher during the WM tasks than they were in the original space. These findings suggested that functional connectivity among the spikes varied dynamically during the WM tasks and could be described effectively in the low dimensional space.
Functional connectivity among spikes in low dimensional space during working memory task in rat.
Ouyang, Mei; Li, Shuangyan; Tian, Xin
2014-01-01
Working memory (WM) is critically important in cognitive tasks. The functional connectivity has been a powerful tool for understanding the mechanism underlying the information processing during WM tasks. The aim of this study is to investigate how to effectively characterize the dynamic variations of the functional connectivity in low dimensional space among the principal components (PCs) which were extracted from the instantaneous firing rate series. Spikes were obtained from medial prefrontal cortex (mPFC) of rats with implanted microelectrode array and then transformed into continuous series via instantaneous firing rate method. Granger causality method is proposed to study the functional connectivity. Then three scalar metrics were applied to identify the changes of the reduced dimensionality functional network during working memory tasks: functional connectivity (GC), global efficiency (E) and casual density (CD). As a comparison, GC, E and CD were also calculated to describe the functional connectivity in the original space. The results showed that these network characteristics dynamically changed during the correct WM tasks. The measure values increased to maximum, and then decreased both in the original and in the reduced dimensionality. Besides, the feature values of the reduced dimensionality were significantly higher during the WM tasks than they were in the original space. These findings suggested that functional connectivity among the spikes varied dynamically during the WM tasks and could be described effectively in the low dimensional space.
Realization of Fractal Affine Transformation
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
This paper gives the definition of fractal affine transformation and presents a specific method for its realization and its cor responding mathematical equations which are essential in fractal image construction.
Fractal Patterns and Chaos Games
Devaney, Robert L.
2004-01-01
Teachers incorporate the chaos game and the concept of a fractal into various areas of the algebra and geometry curriculum. The chaos game approach to fractals provides teachers with an opportunity to help students comprehend the geometry of affine transformations.
Thermodynamics of Fractal Universe
Sheykhi, Ahmad; Wang, Bin
2012-01-01
We investigate the thermodynamical properties of the apparent horizon in a fractal universe. We find that one can always rewrite the Friedmann equation of the fractal universe in the form of the entropy balance relation $ \\delta Q=TdS+Td\\tilde{S}$, where $ \\delta Q $ and $ T $ are the energy flux and Unruh temperature seen by an accelerated observer just inside the apparent horizon, and $d\\tilde{S}$ is the entropy production term due to nonequilibrium thermodynamics of fractal universe. This shows that in a fractal universe, a treatment with nonequilibrium thermodynamics of spacetime may be needed. We also study the generalized second law of thermodynamics in the framework of fractal universe. When the temperature of the apparent horizon and the matter fields inside the horizon are equal, i.e. $T=T_h$, the generalized second law of thermodynamics can be fulfilled provided the deceleration and the equation of state parameters ranges either as $-1 \\leq q < 0 $, $- 1 \\leq w < - 1/3$ or as $q<-1$, $w<...
Ghost quintessence in fractal gravity
Indian Academy of Sciences (India)
Habib Abedi; Mustafa Salti
2015-04-01
In this study, using the time-like fractal theory of gravity, we mainly focus on the ghost dark energy model which was recently suggested to explain the present acceleration of the cosmic expansion. Next, we establish a connection between the quintessence scalar field and fractal ghost dark energy density. This correspondence allows us to reconstruct the potential and the dynamics of a fractal canonical scalar field (the fractal quintessence) according to the evolution of ghost dark energy density.
Research on Fractal-Scanning Path for Arbitrary Boundary Layer in Layered Manufacturing
Institute of Scientific and Technical Information of China (English)
阳佳; 宾鸿赞; 等
2002-01-01
The fractal curve is proposed as a novel scanning-path used in Layered Manufacturing.Aiming at a limitation that the fractal curve can only fill a square region,a method is developed to realize the trimming of fractal curve in arbitrary boundary layer by means of undging intersection points between parameterized arbitrary boundary and a FASS(space-filling,self-avoiding,simple and self-similar)fractal curve.Accordingly,the related algorithm concerning with determining intersection points has been investigated according to the recursion reature of the fractal curve,and in the process of the fractal curve traversed,the rule of udging intersection points is ascertained as well,so that the laser-scanning beam can “walk” along the fractal curve inside the desired boundary,and arbitrary contour components are fabricated.
Scattering in a three-dimensional fuzzy space
Kriel, J. N.; Groenewald, H. W.; Scholtz, F. G.
2017-01-01
We develop scattering theory in a noncommutative space defined by an s u (2 ) coordinate algebra. By introducing a positive operator valued measure as a replacement for strong position measurements, we are able to derive explicit expressions for the probability current, differential and total cross sections. We show that at low incident energies the kinematics of these expressions is identical to that of commutative scattering theory. The consequences of spatial noncommutativity are found to be more pronounced at the dynamical level where, even at low incident energies, the phase shifts of the partial waves can deviate strongly from commutative results. This is demonstrated for scattering from a spherical well. The impact of noncommutativity on the well's spectrum and on the properties of its bound and scattering states are considered in detail. It is found that for sufficiently large well depths the potential effectively becomes repulsive and that the cross section tends towards that of hard sphere scattering. This can occur even at low incident energies when the particle's wavelength inside the well becomes comparable to the noncommutative length scale.
Anisotropic linear elastic properties of fractal-like composites.
Carpinteri, Alberto; Cornetti, Pietro; Pugno, Nicola; Sapora, Alberto
2010-11-01
In this work, the anisotropic linear elastic properties of two-phase composite materials, made up of square inclusions embedded in a matrix, are investigated. The inclusions present a fractal hierarchical distribution and are supposed to have the same Poisson's ratio as the matrix but a different Young's modulus. The effective elastic moduli of the medium are computed at each fractal iteration by coupling a position-space renormalization-group technique with a finite element analysis. The study allows to obtain and generalize some fundamental properties of fractal composite materials.
Anisotropic linear elastic properties of fractal-like composites
Carpinteri, Alberto; Cornetti, Pietro; Pugno, Nicola; Sapora, Alberto
2010-11-01
In this work, the anisotropic linear elastic properties of two-phase composite materials, made up of square inclusions embedded in a matrix, are investigated. The inclusions present a fractal hierarchical distribution and are supposed to have the same Poisson’s ratio as the matrix but a different Young’s modulus. The effective elastic moduli of the medium are computed at each fractal iteration by coupling a position-space renormalization-group technique with a finite element analysis. The study allows to obtain and generalize some fundamental properties of fractal composite materials.
FRACTAL IMAGE FEATURE VECTORS WITH APPLICATIONS IN FRACTOGRAPHY
Directory of Open Access Journals (Sweden)
Hynek Lauschmann
2011-05-01
Full Text Available The morphology of fatigue fracture surface (caused by constant cycle loading is strictly related to crack growth rate. This relation may be expressed, among other methods, by means of fractal analysis. Fractal dimension as a single numerical value is not sufficient. Two types of fractal feature vectors are discussed: multifractal and multiparametric. For analysis of images, the box-counting method for 3D is applied with respect to the non-homogeneity of dimensions (two in space, one in brightness. Examples of application are shown: images of several fracture surfaces are analyzed and related to crack growth rate.
Breban, Romulus
2015-01-01
Five-dimensional (5D) space-time symmetry greatly facilitates how a 4D observer perceives the propagation of a single spinless particle in a 5D space-time. In particular, if the 5D geometry is independent of the fifth coordinate then the 5D physics may be interpreted as 4D quantum mechanics. In this work we address the case where the symmetry is approximate, focusing on the case where the 5D geometry depends weakly on the fifth coordinate. We show that concepts developed for the case of exact...
Fractal geometry and stochastics IV
Bandt, Christoph
2010-01-01
Over the years fractal geometry has established itself as a substantial mathematical theory in its own right. This book collects survey articles covering many of the developments, like Schramm-Loewner evolution, fractal scaling limits, exceptional sets for percolation, and heat kernels on fractals.
Boundary control design for extensible marine risers in three dimensional space
Do, K. D.
2017-02-01
A design of boundary controllers is proposed for (practical) exponential stabilization of extensible marine risers in three-dimensional (3D) space under sea loads. The design removes flaws in existing works. Two Lyapunov-type theorems are developed for study of existence and uniqueness, and stability of nonlinear evolution systems in Hilbert space. These theorems have their potential use in control design and stability analysis for flexible systems including marine risers.
Pair production of Dirac particles in a d+1-dimensional noncommutative space-time
Samary, Dine Ousmane; Hounkonnou, Mahouton Norbert
2014-01-01
This work addresses the exact computation of the propability of fermionic particle pair production in $(d+1)-$ dimensional noncommutative Moyal space. Using the Seiberg-Witten maps that establish relations between noncommutative and commutative field variables, to first order in the noncommutative parameter $\\theta$, we derive the probability density of vacuum-vacuum pair production of Dirac particles. The cases of constant electromagnetic, alternating time-dependent and space-dependent electric fields are considered and discussed.
Pair production of Dirac particles in a -dimensional noncommutative space-time
Ousmane Samary, Dine; N'Dolo, Emanonfi Elias; Hounkonnou, Mahouton Norbert
2014-11-01
This work addresses the computation of the probability of fermionic particle pair production in -dimensional noncommutative Moyal space. Using Seiberg-Witten maps, which establish relations between noncommutative and commutative field variables, up to the first order in the noncommutative parameter , we derive the probability density of vacuum-vacuum pair production of Dirac particles. The cases of constant electromagnetic, alternating time-dependent, and space-dependent electric fields are considered and discussed.
Pair production of Dirac particles in a d + 1-dimensional noncommutative space-time
Energy Technology Data Exchange (ETDEWEB)
Ousmane Samary, Dine [Perimeter Institute for Theoretical Physics, Waterloo, ON (Canada); University of Abomey-Calavi, International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair), Cotonou (Benin); N' Dolo, Emanonfi Elias; Hounkonnou, Mahouton Norbert [University of Abomey-Calavi, International Chair in Mathematical Physics and Applications (ICMPA-UNESCO Chair), Cotonou (Benin)
2014-11-15
This work addresses the computation of the probability of fermionic particle pair production in d + 1-dimensional noncommutative Moyal space. Using Seiberg-Witten maps, which establish relations between noncommutative and commutative field variables, up to the first order in the noncommutative parameter θ, we derive the probability density of vacuum-vacuum pair production of Dirac particles. The cases of constant electromagnetic, alternating time-dependent, and space-dependent electric fields are considered and discussed. (orig.)
Energy Technology Data Exchange (ETDEWEB)
Itoh, S. [Neutron Science Laboratory, Institute of Materials Structure Science, High Energy Accelerator Research Organization, Tsukuba, 305-0801 (Japan)]. E-mail: shinichi.itoh@kek.jp; Kajimoto, R. [Quantum Beam Science Directorate, Japan Atomic Energy Agency, Tokai 319-1195 (Japan); Iwasa, K. [Department of Physics, Tohoku University, Sendai 980-8578 (Japan); Aso, N. [Neutron Science Laboratory, Institute for Solid State Physics, University of Tokyo, Tokai 319-1106 (Japan); Bull, M.J. [ISIS Facility, Rutherford Appleton Laboratory, Didcot, Oxon, OX11 0QX (United Kingdom); Adams, M.A. [ISIS Facility, Rutherford Appleton Laboratory, Didcot, Oxon, OX11 0QX (United Kingdom); Takeuchi, T. [Low Temperature Center, Osaka University, Toyonaka 560-0043 (Japan); Yoshizawa, H. [Neutron Science Laboratory, Institute for Solid State Physics, University of Tokyo, Tokai 319-1106 (Japan)
2006-11-15
We have performed neutron scattering experiments to investigate the static magnetic structure and the magnetic critical scattering in the three-dimensional percolating Heisenberg antiferromagnet, RbMn{sub 0.31}Mg{sub 0.69}F{sub 3}, whose magnetic concentration is just at the percolation concentration for a cubic lattice, c{sub P}=0.312. Magnetic scattering was observed around the superlattice point, (12,12,12). We determined the Neel temperature of this system to be T{sub N}=4.0+/-0.5K, and found that the critical scattering above T{sub N} is similar to that observed in the homogeneous system. At T=1.6K well below T{sub N}, the scattering function is described by a sum of the critical scattering and elastic scattering components, and we demonstrated that the q-variation of the intensity of the elastic scattering component can be described using the fractal dimension.
Harmonic and Dirac oscillators in a (2+1)-dimensional noncommutative space
Vega, F
2013-01-01
We study the Harmonic and Dirac Oscillator problem extended to a three-dimensional noncom- mutative space where the noncommutativity is induced by a shift of the dynamical variables with generators of SL(2;R) in a unitary irreducible representation. The Hilbert space gets the structure of a direct product with the representation space as a factor, where there exist operators which realize the algebra of Lorentz transformations. The spectrum of these models are considered in perturbation theory, both for small and large noncommutativity parameters, ?nding no constraints between coordinates and momenta noncom- mutativity parameters. Since the representation space of the unitary irreducible representations SL(2;R) can be realized in terms of spaces of square-integrable functions, we conclude that these models are equivalent to quantum mechanical models of particles living in a space with an additional compact dimension. PACS: 03.65.-w; 11.30.Cp; 02.40.Gh
Inference and Decoding of Motor Cortex Low-Dimensional Dynamics via Latent State-Space Models.
Aghagolzadeh, Mehdi; Truccolo, Wilson
2016-02-01
Motor cortex neuronal ensemble spiking activity exhibits strong low-dimensional collective dynamics (i.e., coordinated modes of activity) during behavior. Here, we demonstrate that these low-dimensional dynamics, revealed by unsupervised latent state-space models, can provide as accurate or better reconstruction of movement kinematics as direct decoding from the entire recorded ensemble. Ensembles of single neurons were recorded with triple microelectrode arrays (MEAs) implanted in ventral and dorsal premotor (PMv, PMd) and primary motor (M1) cortices while nonhuman primates performed 3-D reach-to-grasp actions. Low-dimensional dynamics were estimated via various types of latent state-space models including, for example, Poisson linear dynamic system (PLDS) models. Decoding from low-dimensional dynamics was implemented via point process and Kalman filters coupled in series. We also examined decoding based on a predictive subsampling of the recorded population. In this case, a supervised greedy procedure selected neuronal subsets that optimized decoding performance. When comparing decoding based on predictive subsampling and latent state-space models, the size of the neuronal subset was set to the same number of latent state dimensions. Overall, our findings suggest that information about naturalistic reach kinematics present in the recorded population is preserved in the inferred low-dimensional motor cortex dynamics. Furthermore, decoding based on unsupervised PLDS models may also outperform previous approaches based on direct decoding from the recorded population or on predictive subsampling.
Coset space dimensional reduction and classification of semi-realistic particle physics models
Energy Technology Data Exchange (ETDEWEB)
Douzas, G.; Grammatikopoulos, T. [National Technical University of Athens, Zografou Campus, 157 80 Zografou, Athens (Greece); Madore, J. [Laboratoire de Physique Theorique, Universite de Paris-Sud, Batiment 211, 91405 Orsay (France); Zoupanos, G.
2008-04-15
Starting from a Yang-Mills-Dirac theory defined in ten dimensions we classify the semi-realistic particle physics models resulting from their Forgacs-Manton dimensional reduction. The higher-dimensional gauge group is chosen to be E{sub 8}. This choice as well as the dimensionality of the space-time is suggested by the heterotic string theory. Furthermore, we assume that the space-time on which the theory is defined can be written in the compactified form M{sup 4} x B, with M{sup 4} the ordinary Minkowski spacetime and B=S/R a 6-dim homogeneous coset space. We constrain our investigation in those cases where the dimensional reduction leads in four dimensions to phenomenologically interesting and anomaly-free GUTs such as E{sub 6}, SO(10) and SU(5). However the four-dimensional surviving scalars transform in the fundamental of the resulting gauge group are not suitable for the superstrong symmetry breaking of the Standard Model. The main objective of our work is the investigation to which extent the latter can be achieved by employing the Wilson flux breaking mechanism. (Abstract Copyright [2008], Wiley Periodicals, Inc.)
Fractal actors and infrastructures
DEFF Research Database (Denmark)
Bøge, Ask Risom
2011-01-01
-network-theory (ANT) into surveillance studies (Ball 2002, Adey 2004, Gad & Lauritsen 2009). In this paper, I further explore the potential of this connection by experimenting with Marilyn Strathern’s concept of the fractal (1991), which has been discussed in newer ANT literature (Law 2002; Law 2004; Jensen 2007). I...... under surveillance. Based on fieldwork conducted in 2008 and 2011 in relation to my Master’s thesis and PhD respectively, I illustrate fractal concepts by describing the acts, actors and infrastructure that make up the ‘DNA surveillance’ conducted by the Danish police....
Deppman, Airton
2016-01-01
The non extensive aspects of $p_T$ distributions obtained in high energy collisions are discussed in relation to possible fractal structure in hadrons, in the sense of the thermofractal structure recently introduced. The evidences of self-similarity in both theoretical and experimental works in High Energy and in Hadron Physics are discussed, to show that the idea of fractal structure of hadrons and fireballs have being under discussion for decades. The non extensive self-consistent thermodynamics and the thermofractal structure allow one to connect non extensivity to intermittence and possibly to parton distribution functions in a single theoretical framework.
Institute of Scientific and Technical Information of China (English)
ZhinhongLi; DongWu; Yuhansun; JunWang; YiLiu; BaozhongDong; Zhinhong
2001-01-01
Silica aggregates were prepared by base-catalyzed hydrolysis and condensation of alkoxides in alcohol.Polyethylene glycol(PEG) was used as organic modifier.The sols were characterized using Small Angle X-ray Scattering (SAXS) with synchrotron radiation as X-ray source.The structure evolution during the sol-gel process was determined and described in terms of the fractal geometry.As-produced silica aggregates were found to be mass fractals.The fractl dimensions spanned the regime 2.1-2.6 corresponding to more branched and compact structures.Both RLCA and Eden models dominated the kinetic growth under base-catalyzed condition.
The Fractal Nature of Wood Revealed by Drying
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
An experiment on wood drying at different temperatures was conducted to show the fractal nature of the pore space within wood. Cubic blocks made from ginkgo (Ginkgo biloba) and Chinese chestnut (Castanea mollissima) wood were used. Samples were dried in oven at the temperature of 20, 40, 60 and 100 ℃, respectively. All the drying procedures lasted four hours. The mass was weighed and the dimensions were measured immediately for each sample when every procedure of drying ended. The fractal dimensions of ...
Large parallel volumes of finite and compact sets in d-dimensional Euclidean space
DEFF Research Database (Denmark)
Kampf, Jürgen; Kiderlen, Markus
The r-parallel volume V (Cr) of a compact subset C in d-dimensional Euclidean space is the volume of the set Cr of all points of Euclidean distance at most r > 0 from C. According to Steiner’s formula, V (Cr) is a polynomial in r when C is convex. For finite sets C satisfying a certain geometric ...
The study of gravitational collapse model in higher dimensional space-time
Debnath, U; Debnath, Ujjal; Chakraborty, Subenoy
2003-01-01
We investigate the end state of the gravitational collapse of an inhomogeneous dust cloud in higher dimensional space-time. The naked singularities are shown to be developing as the final outcome of non-marginally bound collapse. The naked singularities are found to be gravitationally strong in the sense of Tipler .
Collapsing perfect fluid in self-similar five dimensional space-time and cosmic censorship
Ghosh, S G; Saraykar, R V
2014-01-01
We investigate the occurrence and nature of naked singularities in the gravitational collapse of a self-similar adiabatic perfect fluid in a five dimensional space-time. The naked singularities are found to be gravitationally strong in the sense of Tipler and thus violate the cosmic censorship conjecture.
Short-Time Gibbsianness for Infinite-Dimensional Diffusions with Space-Time Interaction
Redig, Frank; Roelly, Sylvie; Ruszel, Wioletta
2010-01-01
We consider a class of infinite-dimensional diffusions where the interaction between the components has a finite extent both in space and time. We start the system from a Gibbs measure with a finite-range uniformly bounded interaction. Under suitable conditions on the drift, we prove that there exis
MATCHING PURSUITS AMONG SHIFTED CAUCHY KERNELS IN HIGHER-DIMENSIONAL SPACES
Institute of Scientific and Technical Information of China (English)
钱涛; 王晋勋; 杨燕
2014-01-01
Appealing to the Clifford analysis and matching pursuits, we study the adaptive decompositions of functions of several variables of finite energy under the dictionaries con-sisting of shifted Cauchy kernels. This is a realization of matching pursuits among shifted Cauchy kernels in higher-dimensional spaces. It offers a method to process signals in arbitrary dimensions.
Faster exact algorithms for computing Steiner trees in higher dimensional Euclidean spaces
DEFF Research Database (Denmark)
Fonseca, Rasmus; Brazil, Marcus; Winter, Pawel;
2016-01-01
The Euclidean Steiner tree problem asks for a network of minimum total length interconnecting a finite set of points in d-dimensional space. For d ≥ 3, only one practical algorithmic approach exists for this problem --- proposed by Smith in 1992. A number of refinements of Smith's algorithm have ...
AN IMPROVED GENETIC ALGORITHM FOR SEARCHING OPTIMAL PARAMETERS IN n—DIMENSIONAL SPACE
Institute of Scientific and Technical Information of China (English)
TangBin; HuGuangrui
2002-01-01
An improved genetic algorithm for searching optimal parameters in n-dimensional space is presented,which encodes movement direction and distance and searches from coarse to precise.The algorithm can realize global optimization and improve the search efficiency,and can be applied effectively in industrial optimization ,data mining and pattern recognition.
AN IMPROVED GENETIC ALGORITHM FOR SEARCHING OPTIMAL PARAMETERS IN n-DIMENSIONAL SPACE
Institute of Scientific and Technical Information of China (English)
Tang Bin; Hu Guangrui
2002-01-01
An improved genetic algorithm for searching optimal parameters in n-dimensional space is presented, which encodes movement direction and distance and searches from coarse to precise. The algorithm can realize global optimization and improve the search efficiency, and can be applied effectively in industrial optimization, data mining and pattern recognition.
Dodge, W. G.
1968-01-01
Computer program determines the forced vibration in three dimensional space of a multiple degree of freedom beam type structural system. Provision is made for the longitudinal axis of the analytical model to change orientation at any point along its length. This program is used by industries in which structural design dynamic analyses are performed.
New Exact Solutions of the Integrable Broer-Kaup Equations in (2+1)-Dimensional Spaces
Institute of Scientific and Technical Information of China (English)
LI De-Sheng; ZHANG Hong-Qing
2004-01-01
In this paper,by improving some procedure of extended tanh-function method,some new exact solutions to the integrable Broer-Kaup equations in(2 + 1)-dimensional spaces are obtained,which include soliton-like solutions,solitary wave solutions,trigonometric function solutions,and rational solutions.
On Schwarzschild black holes in a D-dimensional noncommutative space
Chabab, M; Sedra, M B
2012-01-01
This work aims to implement the idea of noncommutativity in the subject of black holes. Its principal contents deal with a study of Schwarzschild black holes in a D-dimensional noncommutative space. Various aspects related to the non commutative extension are discussed and some non trivial results are derived.
Energy Technology Data Exchange (ETDEWEB)
Marek-Crnjac, L. [Institute of Mathematics and Physics, University of Maribor (Slovenia)], E-mail: Leila.marek@guest.arnes.si; Iovane, G. [DIIMA - Universita di Salerno, Via Ponte don Melillo, 84084 Fisciano (Saudi Arabia) (Italy)], E-mail: iovane@diima.unisa.it; Nada, S.I. [Department of Mathematics, Faculty of Science, Qatar University (Qatar)], E-mail: snada@qu.edu.qa; Zhong, Ting [Department of Mathematics, Jishou University, 427000 Zhangjiajie, Hunan (China)], E-mail: zhongting_2005@126.com
2009-11-30
The present work gives first a review of the mathematical theory of finite and infinite dimensional topological spaces. Subsequently we connect the discussion with E-infinity theory and the theory of partially ordered sets. Finally, we contemplate the relevance of abstract results for quantum gravity.
Coherent States for generalized oscillator with finite-dimensional Hilbert space
Borzov, Vadim V.; Damaskinsky, Eugene V.
2006-01-01
The construction of oscillator-like systems connected with the given set of orthogonal polynomials and coherent states for such systems developed by authors is extended to the case of the systems with finite-dimensional state space. As example we consider the generalized oscillator connected with Krawtchouk polynomials.
Numerical Method for Following the Closed Orbits in High-dimensional Space
Institute of Scientific and Technical Information of China (English)
武际可; 周鵾
1994-01-01
The spline interpolation is used to approximate the closed orbits, and the problem of following the dosed orbit in large-scale is turned to tracking the solution curve of a nonlinear equation system in higher-dimensional space. The deformation of the closed orbit of Lorenz equation is calculated.
Integrability of Nonlinear Equations of Motion on Two-Dimensional World Sheet Space-Time
Institute of Scientific and Technical Information of China (English)
YAN Jun
2005-01-01
The integrability character of nonlinear equations of motion of two-dimensional gravity with dynamical torsion and bosonic string coupling is studied in this paper. The space-like and time-like first integrals of equations of motion are also found.
Translation surfaces in the three-dimensional simply isotropic space 𝕀31
Karacan, Murat Kemal; Yoon, Dae Won; Bukcu, Bahaddin
2016-05-01
In this paper, we classify translation surfaces in the three-dimensional simply isotropic space 𝕀31 under the condition Δix i = λixi where Δ is the Laplace operator with respect to the first and second fundamental forms and λ is a real number. We also give explicit forms of these surfaces.
T.A. Knoch (Tobias)
2000-01-01
textabstractDespite the successful linear sequencing of the human genome its three-dimensional structure is widely unknown, although it is important for gene regulation and replication. For a long time the interphase nucleus has been viewed as a 'spaghetti soup' of DNA without much internal stru
T.A. Knoch (Tobias)
2000-01-01
textabstractDespite the successful linear sequencing of the human genome its three-dimensional structure is widely unknown, although it is important for gene regulation and replication. For a long time the interphase nucleus has been viewed as a 'spaghetti soup' of DNA without much internal struc
Fractal elements and their applications
Gil’mutdinov, Anis Kharisovich; El-Khazali, Reyad
2017-01-01
This book describes a new type of passive electronic components, called fractal elements, from a theoretical and practical point of view. The authors discuss in detail the physical implementation and design of fractal devices for application in fractional-order signal processing and systems. The concepts of fractals and fractal signals are explained, as well as the fundamentals of fractional calculus. Several implementations of fractional impedances are discussed, along with comparison of their performance characteristics. Details of design, schematics, fundamental techniques and implementation of RC-based fractal elements are provided. .
Stretched Exponential Relaxation in Disordered Complex Systems: Fractal Time Random Walk Model
Institute of Scientific and Technical Information of China (English)
Ekrem Aydmer
2007-01-01
We have analytically derived the relaxation function for one-dimensional disordered complex systems in terms of autocorrelation function of fractal time random walk by using operator formalism. We have shown that the relaxation function has stretched exponential, i.e. the Kohlrausch-Williams-Watts character for a fractal time random walk process.
Institute of Scientific and Technical Information of China (English)
张庆顺; 马跃峰
2013-01-01
The fractal eco-adaptation and isomorphic wholeness of the spatial form of the traditional mountain settlement were formed gradually due to the self-organization and self-adaptation of the complex system of the settlement. Fractal topological structure between interior space and exterior space, and the exterior space' s fractal aesthetic features are explored, where the author points out that fractal adaptation and artistic integration of natural ecology, architectural form and humanity ecology have resulted in traditional mountain settlement' s exterior spatial gestalt structure and holistic order. Taking into consideration of phenomenological theory on spatial perception, the author finally argues that the experience ot exterior spatial order of the traditional mountain settlement is one of fractal space-time and coexistence of chaos and order.%首先提出传统山地聚落因其复杂系统的自组织与自适应,逐渐形成空间形态分形的生态适应性和同构的整体性;进而进行聚落内外空间分形的拓扑结构研究,探讨外部空间的分形美学特征,指出生态、形态、文态的分形适应和艺术整合形成了传统山地聚落外部空间的完形结构和整体秩序；最后结合现象学的空间知觉理论,指出传统山地聚落外部空间的体验既是时空分形的体验,又是混沌与秩序并存的体验.
Fractal frontiers in cardiovascular magnetic resonance: towards clinical implementation.
Captur, Gabriella; Karperien, Audrey L; Li, Chunming; Zemrak, Filip; Tobon-Gomez, Catalina; Gao, Xuexin; Bluemke, David A; Elliott, Perry M; Petersen, Steffen E; Moon, James C
2015-09-07
Many of the structures and parameters that are detected, measured and reported in cardiovascular magnetic resonance (CMR) have at least some properties that are fractal, meaning complex and self-similar at different scales. To date however, there has been little use of fractal geometry in CMR; by comparison, many more applications of fractal analysis have been published in MR imaging of the brain.This review explains the fundamental principles of fractal geometry, places the fractal dimension into a meaningful context within the realms of Euclidean and topological space, and defines its role in digital image processing. It summarises the basic mathematics, highlights strengths and potential limitations of its application to biomedical imaging, shows key current examples and suggests a simple route for its successful clinical implementation by the CMR community.By simplifying some of the more abstract concepts of deterministic fractals, this review invites CMR scientists (clinicians, technologists, physicists) to experiment with fractal analysis as a means of developing the next generation of intelligent quantitative cardiac imaging tools.
Fractal Representation of Exergy
Directory of Open Access Journals (Sweden)
Yvain Canivet
2016-02-01
Full Text Available We developed a geometrical model to represent the thermodynamic concepts of exergy and anergy. The model leads to multi-scale energy lines (correlons that we characterised by fractal dimension and entropy analyses. A specific attention will be paid to overlapping points, rising interesting remarks about trans-scale dynamics of heat flows.
Pelletier, J D
1997-01-01
The power spectrum S of linear transects of the earth's topography is often observed to be a power-law function of wave number k with exponent close to -2: S(k) is proportional to k^-2. In addition, river networks are fractal trees that satisfy many power-law or fractal relationships between their morphologic components. A model equation for the evolution of the earth's topography by erosional processes which produces fractal topography and fractal river networks is presented and its solutions compared in detail to real topography. The model is the diffusion equation for sediment transport on hillslopes and channels with the local diffusivity proportional to the square of the discharge. The dependence of diffusivity on discharge follows from fundamental equations of sediment transport. We study the model in two ways. In the first analysis the diffusivity is parameterized as a function of relief and a Taylor expansion procedure is carried out to obtain a differential equation for the landform elevation which i...
Earnshow, R; Jones, H
1991-01-01
This volume is based upon the presentations made at an international conference in London on the subject of 'Fractals and Chaos'. The objective of the conference was to bring together some of the leading practitioners and exponents in the overlapping fields of fractal geometry and chaos theory, with a view to exploring some of the relationships between the two domains. Based on this initial conference and subsequent exchanges between the editors and the authors, revised and updated papers were produced. These papers are contained in the present volume. We thank all those who contributed to this effort by way of planning and organisation, and also all those who helped in the production of this volume. In particular, we wish to express our appreciation to Gerhard Rossbach, Computer Science Editor, Craig Van Dyck, Production Director, and Nancy A. Rogers, who did the typesetting. A. J. Crilly R. A. Earnshaw H. Jones 1 March 1990 Introduction Fractals and Chaos The word 'fractal' was coined by Benoit Mandelbrot i...
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.
ON THE GAUSS MAP OF RULED SURFACES OF TYPE II IN 3-DIMENSIONAL PSEUDO-GALILEAN SPACE
Directory of Open Access Journals (Sweden)
Alper Osman Öğrenmis
2013-02-01
Full Text Available In this paper, ruled surfaces of type II in a three-dimensional Pseudo-Galilean space are given. By studying its Gauss map and Laplacian operator, we obtain a classification of ruled surfaces of type II in a three-dimensional Pseudo-Galilean space.
Numerical relativity for D dimensional axially symmetric space-times: formalism and code tests
Zilhao, Miguel; Sperhake, Ulrich; Cardoso, Vitor; Gualtieri, Leonardo; Herdeiro, Carlos; Nerozzi, Andrea
2010-01-01
The numerical evolution of Einstein's field equations in a generic background has the potential to answer a variety of important questions in physics: from applications to the gauge-gravity duality, to modelling black hole production in TeV gravity scenarios, analysis of the stability of exact solutions and tests of Cosmic Censorship. In order to investigate these questions, we extend numerical relativity to more general space-times than those investigated hitherto, by developing a framework to study the numerical evolution of D dimensional vacuum space-times with an SO(D-2) isometry group for D\\ge 5, or SO(D-3) for D\\ge 6. Performing a dimensional reduction on a (D-4)-sphere, the D dimensional vacuum Einstein equations are rewritten as a 3+1 dimensional system with source terms, and presented in the Baumgarte, Shapiro, Shibata and Nakamura (BSSN) formulation. This allows the use of existing 3+1 dimensional numerical codes with small adaptations. Brill-Lindquist initial data are constructed in D dimensions an...
Directory of Open Access Journals (Sweden)
BACCHI O.O.S.
1996-01-01
Full Text Available Fractal scaling has been applied to soils, both for void and solid phases, as an approach to characterize the porous arrangement, attempting to relate particle-size distribution to soil water retention and soil water dynamic properties. One important point of such an analysis is the assumption that the void space geometry of soils reflects its solid phase geometry, taking into account that soil pores are lined by the full range of particles, and that their fractal dimension, which expresses their tortuosity, could be evaluated by the fractal scaling of particle-size distribution. Other authors already concluded that although fractal scaling plays an important role in soil water retention and porosity, particle-size distribution alone is not sufficient to evaluate the fractal structure of porosity. It is also recommended to examine the relationship between fractal properties of solids and of voids, and in some special cases, look for an equivalence of both fractal dimensions. In the present paper data of 42 soil samples were analyzed in order to compare fractal dimensions of pore-size distribution, evaluated by soil water retention curves (SWRC of soils, with fractal dimensions of soil particle-size distributions (PSD, taking the hydraulic conductivity as a standard variable for the comparison, due to its relation to tortuosity. A new procedure is proposed to evaluate the fractal dimension of pore-size distribution. Results indicate a better correlation between fractal dimensions of pore-size distribution and the hydraulic conductivity for this set of soils, showing that for most of the soils analyzed there is no equivalence of both fractal dimensions. For most of these soils the fractal dimension of particle-size distribution does not indicate properly the pore trace tortuosity. A better equivalence of both fractal dimensions was found for sandy soils.
National Research Council Canada - National Science Library
Ucar, Murat; Guryildirim, Melike; Tokgoz, Nil; Kilic, Koray; Borcek, Alp; Oner, Yusuf; Akkan, Koray; Tali, Turgut
2014-01-01
...) high-sampling-efficiency technique (sampling perfection with application optimized contrast using different flip angle evolutions [SPACE]) and T2-weighted (T2W) two-dimensional (2D) turbo spin echo (TSE...
Numerical relativity for D dimensional axially symmetric space-times: Formalism and code tests
Zilhão, Miguel; Witek, Helvi; Sperhake, Ulrich; Cardoso, Vitor; Gualtieri, Leonardo; Herdeiro, Carlos; Nerozzi, Andrea
2010-04-01
The numerical evolution of Einstein’s field equations in a generic background has the potential to answer a variety of important questions in physics: from applications to the gauge-gravity duality, to modeling black hole production in TeV gravity scenarios, to analysis of the stability of exact solutions, and to tests of cosmic censorship. In order to investigate these questions, we extend numerical relativity to more general space-times than those investigated hitherto, by developing a framework to study the numerical evolution of D dimensional vacuum space-times with an SO(D-2) isometry group for D≥5, or SO(D-3) for D≥6. Performing a dimensional reduction on a (D-4) sphere, the D dimensional vacuum Einstein equations are rewritten as a 3+1 dimensional system with source terms, and presented in the Baumgarte, Shapiro, Shibata, and Nakamura formulation. This allows the use of existing 3+1 dimensional numerical codes with small adaptations. Brill-Lindquist initial data are constructed in D dimensions and a procedure to match them to our 3+1 dimensional evolution equations is given. We have implemented our framework by adapting the Lean code and perform a variety of simulations of nonspinning black hole space-times. Specifically, we present a modified moving puncture gauge, which facilitates long-term stable simulations in D=5. We further demonstrate the internal consistency of the code by studying convergence and comparing numerical versus analytic results in the case of geodesic slicing for D=5, 6.
On the permeability of fractal tube bundles
Zinovik, I
2011-01-01
The permeability of a porous medium is strongly affected by its local geometry and connectivity, the size distribution of the solid inclusions and the pores available for flow. Since direct measurements of the permeability are time consuming and require experiments that are not always possible, the reliable theoretical assessment of the permeability based on the medium structural characteristics alone is of importance. When the porosity approaches unity, the permeability-porosity relationships represented by the Kozeny-Carman equations and Archie's law predict that permeability tends to infinity and thus they yield unrealistic results if specific area of the porous media does not tend to zero. The goal of this paper is an evaluation of the relationships between porosity and permeability for a set of fractal models with porosity approaching unity and a finite permeability. It is shown that the two-dimensional foams generated by finite iterations of the corresponding geometric fractals can be used to model poro...
The Extended Loop Group An Infinite Dimensional Manifold Associated with the Loop Space
Di Bartolo, C; Griego, J R; Bartolo, Cayetano Di; Gambini, Rodolfo; Griego, Jorge
1993-01-01
A set of coordinates in the non parametric loop-space is introduced. We show that these coordinates transform under infinite dimensional linear representations of the diffeomorphism group. An extension of the group of loops in terms of these objects is proposed. The enlarged group behaves locally as an infinite dimensional Lie group. Ordinary loops form a subgroup of this group. The algebraic properties of this new mathematical structure are analized in detail. Applications of the formalism to field theory, quantum gravity and knot theory are considered.
Examples of bosonic de Finetti states over finite dimensional Hilbert spaces
Gottlieb, A D
2005-01-01
According to the Quantum de Finetti Theorem, locally normal infinite particle states with Bose-Einstein symmetry can be represented as mixtures of infinite tensor powers of vector states. This note presents examples of infinite-particle states with Bose-Einstein symmetry that arise as limits of Gibbs ensembles on finite dimensional spaces, and displays their de Finetti representations. We consider Gibbs ensembles for systems of bosons in a finite dimensional setting and discover limits as the number of particles tends to infinity, provided the temperature is scaled in proportion to particle number.
Three-dimensional SCFT on conic space as hologram of charged topological black hole
Energy Technology Data Exchange (ETDEWEB)
Huang, Xing [School of Physics & Astronomy and Center for Theoretical Physics,Seoul National University, Seoul 151-747 (Korea, Republic of); Rey, Soo-Jong [School of Physics & Astronomy and Center for Theoretical Physics, Seoul National University, Seoul 151-747 (Korea, Republic of); Center for Quantum Space-Time, Sogang University,Seoul 121-742 (Korea, Republic of); Zhou, Yang [Center for Quantum Space-Time, Sogang University, Seoul 121-742 (Korea, Republic of)
2014-03-26
We construct three-dimensional N=2 supersymmetric field theories on conic spaces. Built upon the fact that the partition function depends solely on the Reeb vector of the Killing vector, we propose that holographic dual of these theories are four-dimensional, supersymmetric charged topological black holes. With the supersymmetry localization technique, we study conserved supercharges, free energy, and supersymmetric Rényi entropy. At planar large N limit, we demonstrate perfect agreement between the superconformal field theories and the supersymmetric charged topological black holes.
Huang, Xing; Zhou, Yang
2014-01-01
We construct three-dimensional N=2 supersymmetric conformal field theories on conic spaces. Built upon the fact that the partition function depends solely on the Reeb vector of the Killing vector, we propose that holographic dual of these theories are four-dimensional, supersymmetric charged topological black holes. With the supersymmetry localization technique, we study conserved supercharges, free energy, and Renyi entropy. At planar large N limit, we demonstrate perfect agreement between the superconformal field theories and the supersymmetric charged topological black holes.
Discrimination and synthesis of recursive quantum states in high-dimensional Hilbert spaces
Simon, David S.; Fitzpatrick, Casey A.; Sergienko, Alexander V.
2015-04-01
We propose an interferometric method for statistically discriminating between nonorthogonal states in high-dimensional Hilbert spaces for use in quantum information processing. The method is illustrated for the case of photon orbital angular momentum (OAM) states. These states belong to pairs of bases that are mutually unbiased on a sequence of two-dimensional subspaces of the full Hilbert space, but the vectors within the same basis are not necessarily orthogonal to each other. Over multiple trials, this method allows distinguishing OAM eigenstates from superpositions of multiple such eigenstates. Variations of the same method are then shown to be capable of preparing and detecting arbitrary linear combinations of states in Hilbert space. One further variation allows the construction of chains of states obeying recurrence relations on the Hilbert space itself, opening a new range of possibilities for more abstract information-coding algorithms to be carried out experimentally in a simple manner. Among other applications, we show that this approach provides a simplified means of switching between pairs of high-dimensional mutually unbiased OAM bases.
We live in the quantum 4-dimensional Minkowski space-time
Hwang, W-Y Pauchy
2015-01-01
We try to define "our world" by stating that "we live in the quantum 4-dimensional Minkowski space-time with the force-fields gauge group $SU_c(3) \\times SU_L(2) \\times U(1) \\times SU_f(3)$ built-in from the outset". We begin by explaining what "space" and "time" are meaning for us - the 4-dimensional Minkowski space-time, then proceeding to the quantum 4-dimensional Minkowski space-time. In our world, there are fields, or, point-like particles. Particle physics is described by the so-called Standard Model. Maybe I should explain why, how, and what my Standard Model would be everybody's "Standard Model" some day. Following the thinking underlying the minimal Standard Model and based on the gauge group $SU_c(3) \\times SU_L(2) \\times U(1) \\times SU_f(3)$, the extension, which is rather unique, derives from the family concept that there are three generations of quarks, on (123), and of leptons, on another (123). It yields neutrino oscillations in a natural manner. It also predicts a variety of lepton-flavor-viol...
Fractal Dimension as a measure of the scale of Homogeneity
Yadav, Jaswant K; Khandai, Nishikanta
2010-01-01
In the multi-fractal analysis of large scale matter distribution, the scale of transition to homogeneity is defined as the scale above which the fractal dimension of underlying point distribution is equal to the ambient dimension of the space in which points are distributed. With finite sized weakly clustered distribution of tracers obtained from galaxy redshift surveys it is difficult to achieve this equality. Recently we have defined the scale of homogeneity to be the scale above which the deviation of fractal dimension from the ambient dimension becomes smaller than the statistical dispersion. In this paper we use the relation between the fractal dimensions and the correlation function to compute the dispersion for any given model in the limit of weak clustering amplitude. We compare the deviation and dispersion for the LCDM model and discuss the implication of this comparison for the expected scale of homogeneity in the concordant model of cosmology. We estimate the upper limit to the scale of homogeneity...
A variable-order fractal derivative model for anomalous diffusion
Directory of Open Access Journals (Sweden)
Liu Xiaoting
2017-01-01
Full Text Available This paper pays attention to develop a variable-order fractal derivative model for anomalous diffusion. Previous investigations have indicated that the medium structure, fractal dimension or porosity may change with time or space during solute transport processes, results in time or spatial dependent anomalous diffusion phenomena. Hereby, this study makes an attempt to introduce a variable-order fractal derivative diffusion model, in which the index of fractal derivative depends on temporal moment or spatial position, to characterize the above mentioned anomalous diffusion (or transport processes. Compared with other models, the main advantages in description and the physical explanation of new model are explored by numerical simulation. Further discussions on the dissimilitude such as computational efficiency, diffusion behavior and heavy tail phenomena of the new model and variable-order fractional derivative model are also offered.
A turbulent premixed flame on fractal-grid generated turbulence
Soulopoulos, Nikos; Beyrau, Frank; Hardalupas, Yannis; Taylor, A M K P; Vassilicos, J Christos
2010-01-01
A space-filling, low blockage fractal grid is used as a novel turbulence generator in a premixed turbulent combustion experiment. In contrast to the power law decay of a standard turbulence grid, the downstream turbulence intensity of the fractal grid increases until it reaches a peak at some distance from the grid before it finally decays. The effective mesh size and the solidity are the same as those of a standard square mesh grid with which it is compared. It is found that, for the same flow rate and stoichiometry, the fractal generated turbulence enhances the burning rate and causes the flame to further increase its area. Using a flame fractal model, an attempt is made to highlight differences between the flames established at the two different turbulent fields.
Taylor, Shawn C.
2011-01-01
A noncontacting, two-dimensional (2-D) laser inspection system has been designed and implemented to dimensionally profile thermal barriers being developed for space vehicle applications. In a vehicle as-installed state, thermal barriers are commonly compressed between load sensitive thermal protection system (TPS) panels to prevent hot gas ingestion through the panel interface during flight. Loads required to compress the thermal barriers are functions of their construction, as well as their dimensional characteristics relative to the gaps in which they are installed. Excessive loads during a mission could damage surrounding TPS panels and have catastrophic consequences. As such, accurate dimensional profiling of thermal barriers prior to use is important. Due to the compliant nature of the thermal barriers, traditional contact measurement techniques (e.g., calipers and micrometers) are subjective and introduce significant error and variability into collected dimensional data. Implementation of a laser inspection system significantly enhanced the method by which thermal barriers are dimensionally profiled, and improved the accuracy and repeatability of collected data. A statistical design of experiments study comparing laser inspection and manual caliper measurement techniques verified these findings.
Fractality Field in the Theory of Scale Relativity
Directory of Open Access Journals (Sweden)
Nottale L.
2005-04-01
Full Text Available In the theory of scale relativity, space-time is considered to be a continuum that is not only curved, but also non-differentiable, and, as a consequence, fractal. The equation of geodesics in such a space-time can be integrated in terms of quantum mechanical equations. We show in this paper that the quantum potential is a manifestation of such a fractality of space-time (in analogy with Newton’s potential being a manifestation of curvature in the framework of general relativity.
The Loeb Space of Denumerable Infinite Dimensional Probability Product Measure Spaces
Institute of Scientific and Technical Information of China (English)
陈东立
2003-01-01
Let {(Xi, Si, μi) : i ∈N} be a sequence of probability measure spaces and (↑*Xi, L(↑*Si), L(↑*μi)) be the Loeb measure space with respect to (Xi, Si, μi)for i∈N.lET x=↑∞×i=1 Xi,S=↑∞×i=1 Si,μ=↑∞×i=1 μi,We prove that ↑∞×i=1 L(↑*Si)（∪→）L1(↑*L)and L(↑*μ）|↑∞×i=L(↑*Si)=L(↑*μi) in embedding meaning.
Vos, Gordon A.; Fink, Patrick; Ngo, Phong H.; Morency, Richard; Simon, Cory; Williams, Robert E.; Perez, Lance C.
2017-01-01
Space Human Factors and Habitability (SHFH) Element within the Human Research Program (HRP) and the Behavioral Health and Performance (BHP) Element are conducting research regarding Net Habitable Volume (NHV), the internal volume within a spacecraft or habitat that is available to crew for required activities, as well as layout and accommodations within the volume. NASA needs methods to unobtrusively collect NHV data without impacting crew time. Data required includes metrics such as location and orientation of crew, volume used to complete tasks, internal translation paths, flow of work, and task completion times. In less constrained environments methods exist yet many are obtrusive and require significant post-processing. ?Examplesused in terrestrial settings include infrared (IR) retro-reflective marker based motion capture, GPS sensor tracking, inertial tracking, and multi-camera methods ?Due to constraints of space operations many such methods are infeasible. Inertial tracking systems typically rely upon a gravity vector to normalize sensor readings,and traditional IR systems are large and require extensive calibration. ?However, multiple technologies have not been applied to space operations for these purposes. Two of these include: 3D Radio Frequency Identification Real-Time Localization Systems (3D RFID-RTLS) ?Depth imaging systems which allow for 3D motion capture and volumetric scanning (such as those using IR-depth cameras like the Microsoft Kinect or Light Detection and Ranging / Light-Radar systems, referred to as LIDAR)
Hofstadter Butterfly Diagram in Noncommutative Space
Takahashi, H; Takahashi, Hidenori; Yamanaka, Masanori
2006-01-01
We study an energy spectrum of electron moving under the constant magnetic field in two dimensional noncommutative space. It take place with the gauge invariant way. The Hofstadter butterfly diagram of the noncommutative space is calculated in terms of the lattice model which is derived by the Bopp's shift for space and by the Peierls substitution for external magnetic field. We also find the fractal structure in new diagram. Although the global features of the new diagram are similar to the diagram of the commutative space, the detail structure is different from it.
Energy Technology Data Exchange (ETDEWEB)
Breban, Romulus [Institut Pasteur, Paris Cedex 15 (France)
2016-09-15
Five-dimensional (5D) space-time symmetry greatly facilitates how a 4D observer perceives the propagation of a single spinless particle in a 5D space-time. In particular, if the 5D geometry is independent of the fifth coordinate then the 5D physics may be interpreted as 4D quantum mechanics. In this work we address the case where the symmetry is approximate, focusing on the case where the 5D geometry depends weakly on the fifth coordinate. We show that concepts developed for the case of exact symmetry approximately hold when other concepts such as decaying quantum states, resonant quantum scattering, and Stokes drag are adopted, as well. We briefly comment on the optical model of the nuclear interactions and Millikan's oil drop experiment. (orig.)
Breban, Romulus
2016-09-01
Five-dimensional (5D) space-time symmetry greatly facilitates how a 4D observer perceives the propagation of a single spinless particle in a 5D space-time. In particular, if the 5D geometry is independent of the fifth coordinate then the 5D physics may be interpreted as 4D quantum mechanics. In this work we address the case where the symmetry is approximate, focusing on the case where the 5D geometry depends weakly on the fifth coordinate. We show that concepts developed for the case of exact symmetry approximately hold when other concepts such as decaying quantum states, resonant quantum scattering, and Stokes drag are adopted, as well. We briefly comment on the optical model of the nuclear interactions and Millikan's oil drop experiment.
Low dimensional representation of face space by face-selective inferior temporal neurons.
Salehi, Sina; Dehaqani, Mohammad-Reza A; Esteky, Hossein
2017-05-01
The representation of visual objects in primate brain is distributed and multiple neurons are involved in encoding each object. One way to understand the neural basis of object representation is to estimate the number of neural dimensions that are needed for veridical representation of object categories. In this study, the characteristics of the match between physical-shape and neural representational spaces in monkey inferior temporal (IT) cortex were evaluated. Specifically, we examined how the number of neural dimensions, stimulus behavioral saliency and stimulus category selectivity of neurons affected the correlation between shape and neural representational spaces in IT cortex. Single-unit recordings from monkey IT cortex revealed that there was a significant match between face space and its neural representation at lower neural dimensions, whereas the optimal match for the non-face objects was observed at higher neural dimensions. There was a statistically significant match between the face and neural spaces only in the face-selective neurons, whereas a significant match was observed for non-face objects in all neurons regardless of their category selectivity. Interestingly, the face neurons showed a higher match for the non-face objects than for the faces at higher neural dimensions. The optimal representation of face space in the responses of the face neurons was a low dimensional map that emerged early (~150 ms post-stimulus onset) and was followed by a high dimensional and relatively late (~300 ms) map for the non-face stimuli. These results support a multiplexing function for the face neurons in the representation of very similar shape spaces, but with different dimensionality and timing scales. © 2017 Federation of European Neuroscience Societies and John Wiley & Sons Ltd.
Fractal and transfractal recursive scale-free nets
Energy Technology Data Exchange (ETDEWEB)
Rozenfeld, Hernan D [Department of Physics, Clarkson University, Potsdam, NY 13699-5820 (United States); Havlin, Shlomo [Minerva Center and Department of Physics, Bar-Ilan University, Ramat-Gan 52900 (Israel); Ben-Avraham, Daniel [Department of Physics, Clarkson University, Potsdam, NY 13699-5820 (United States)
2007-06-15
We explore the concepts of self-similarity, dimensionality, and (multi)scaling in a new family of recursive scale-free nets that yield themselves to exact analysis through renormalization techniques. All nets in this family are self-similar and some are fractals-possessing a finite fractal dimension-while others are small-world (their diameter grows logarithmically with their size) and are infinite-dimensional. We show how a useful measure of transfinite dimension may be defined and applied to the small-world nets. Concerning multiscaling, we show how first-passage time for diffusion and resistance between hubs (the most connected nodes) scale differently than for other nodes. Despite the different scalings, the Einstein relation between diffusion and conductivity holds separately for hubs and nodes. The transfinite exponents of small-world nets obey Einstein relations analogous to those in fractal nets.
Fermion confinement via quantum walks in (2+1)-dimensional and (3+1)-dimensional space-time
Márquez-Martín, I.; Di Molfetta, G.; Pérez, A.
2017-04-01
We analyze the properties of a two- and three-dimensional quantum walk that are inspired by the idea of a brane-world model put forward by Rubakov and Shaposhnikov [Phys. Lett. B 125, 136 (1983), 10.1016/0370-2693(83)91253-4]. In that model, particles are dynamically confined on the brane due to the interaction with a scalar field. We translated this model into an alternate quantum walk with a coin that depends on the external field, with a dependence which mimics a domain wall solution. As in the original model, fermions (in our case, the walker) become localized in one of the dimensions, not from the action of a random noise on the lattice (as in the case of Anderson localization) but from a regular dependence in space. On the other hand, the resulting quantum walk can move freely along the "ordinary" dimensions.
Energy Technology Data Exchange (ETDEWEB)
Bhar, Piyali; Rahaman, Farook [Jadavpur University, Department of Mathematics, Kolkata, West Bengal (India)
2014-12-01
In this paper we ask whether the wormhole solutions exist in different dimensional noncommutativity-inspired spacetimes. It is well known that the noncommutativity of the space is an outcome of string theory and it replaced the usual point-like object by a smeared object. Here we have chosen the Lorentzian distribution as the density function in the noncommutativity-inspired spacetime. We have observed that the wormhole solutions exist only in four and five dimensions; however, in higher than five dimensions no wormhole exists. For five dimensional spacetime, we get a wormhole for a restricted region. In the usual four dimensional spacetime, we get a stable wormhole which is asymptotically flat. (orig.)
Higher-dimensional unification with continuous and fuzzy coset spaces as extra dimensions
Energy Technology Data Exchange (ETDEWEB)
Gavriil, D.; Manolakos, G.; Orfanidis, G. [Physics Department, National Technical University, Athens (Greece); Zoupanos, G. [Physics Department, National Technical University, Athens (Greece); Max-Planck-Institut fuer Physik (Werner-Heisenberg-Institut), Muenchen (Germany); Arnold-Sommerfeld-Center fuer Theoretische Physik Department fuer Physik, Ludwig-Maximilians-Universitaet Muenchen (Germany)
2015-07-15
We first review the Coset Space Dimensional Reduction (CSDR) programme and present the best model constructed so far based on the N = 1, 10-dimensional E{sub 8} gauge theory reduced over the nearly-Kaehler manifold SU(3) / U(1) x U(1) with the additional use of the Wilson flux mechanism. Then we present the corresponding programme in the case that the extra dimensions are considered to be fuzzy coset spaces and the best model that has been constructed in this framework too. In both cases the best model appears to be the trinification GUT SU(3){sup 3}. (copyright 2015 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
Transformation of general astigmatic Gaussian beams in a four-dimensional phase space
Institute of Scientific and Technical Information of China (English)
Baoxin Chen
2006-01-01
@@ A phase space model of two-dimensional (2D) Gaussian beam propagation is generalized for threedimensional (3D) general astigmatic Gaussian beam passing through first-order optical system. The general astigmatic Gaussian beam is represented by a four-dimensional (4D) phase super-ellipsoid that defined by an associated 4 × 4 real matrix, then the transformation formula of the phase super-ellipsoid of the beam through first-order optical system is derived. In particular, in the phase space framework, the beam propagation factor M2 value is proved to be a ratio of phase area of real beam to ideal beam, and a novel approach for a qualitative examination of the properties of fractional Fourier transform (FRT) for the beam is also provided.
Distributed spirals: a new class of three-dimensional k-space trajectories.
Turley, Dallas C; Pipe, James G
2013-08-01
This work presents a new class of three-dimensional spiral based-trajectories for sampling magnetic resonance data. The distributed spirals trajectory efficiently traverses a cylinder or sphere or intermediate shape in k-space. The trajectory is shown to be nearly as efficient as a conventional stack of spirals trajectory in terms of scan time and signal-to-noise ratio, while reducing coherent aliasing in all three spatial directions and reducing Gibbs ringing due to the nature of collecting data from a sphere in k-space. The trajectory uses a single two-dimensional spiral waveform with the addition of a single orthogonal waveform which is scaled with each repetition, making it relatively easy to implement. Blurring from off-resonance only occurs in two dimensions due to the temporal nature of the sampling.
Particle Filtering for Large Dimensional State Spaces with Multimodal Observation Likelihoods
Vaswani, Namrata
2008-01-01
We study efficient importance sampling techniques for particle filtering (PF) when either (a) the observation likelihood (OL) is frequently multimodal or heavy-tailed, or (b) the state space dimension is large or both. When the OL is multimodal, but the state transition pdf (STP) is narrow enough, the optimal importance density is usually unimodal. Under this assumption, many techniques have been proposed. But when the STP is broad, this assumption does not hold. We study how existing techniques can be generalized to situations where the optimal importance density is multimodal, but is unimodal conditioned on a part of the state vector. Sufficient conditions to test for the unimodality of this conditional posterior are derived. The number of particles, N, to accurately track using a PF increases with state space dimension, thus making any regular PF impractical for large dimensional tracking problems. We propose a solution that partially addresses this problem. An important class of large dimensional problems...
Hybrid-space density matrix renormalization group study of the doped two-dimensional Hubbard model
Ehlers, G.; White, S. R.; Noack, R. M.
2017-03-01
The performance of the density matrix renormalization group (DMRG) is strongly influenced by the choice of the local basis of the underlying physical lattice. We demonstrate that, for the two-dimensional Hubbard model, the hybrid-real-momentum-space formulation of the DMRG is computationally more efficient than the standard real-space formulation. In particular, we show that the computational cost for fixed bond dimension of the hybrid-space DMRG is approximately independent of the width of the lattice, in contrast to the real-space DMRG, for which it is proportional to the width squared. We apply the hybrid-space algorithm to calculate the ground state of the doped two-dimensional Hubbard model on cylinders of width four and six sites; at n =0.875 filling, the ground state exhibits a striped charge-density distribution with a wavelength of eight sites for both U /t =4.0 and 8.0 . We find that the strength of the charge ordering depends on U /t and on the boundary conditions. Furthermore, we investigate the magnetic ordering as well as the decay of the static spin, charge, and pair-field correlation functions.
Eigenmodes of 3-dimensional spherical spaces and their application to cosmology
Lehoucq, R; Uzan, J P; Gausmann, E; Luminet, Jean Pierre; Lehoucq, Roland; Weeks, Jeffrey; Uzan, Jean-Philippe; Gausmann, Evelise; Luminet, Jean-Pierre
2002-01-01
This article investigates the computation of the eigenmodes of the Laplacian operator in multi-connected three-dimensional spherical spaces. General mathematical results and analytical solutions for lens and prism spaces are presented. Three complementary numerical methods are developed and compared with our analytic results and previous investigations. The cosmological applications of these results are discussed, focusing on the cosmic microwave background (CMB) anisotropies. In particular, whereas in the Euclidean case too small universes are excluded by present CMB data, in the spherical case there will always exist candidate topologies even if the total energy density parameter of the universe is very close to unity.
On the time-dependent extra spatial dimensions in six dimensional space-time
Lien, Phan Hong; Hai, Do Thi Hong
2016-06-01
In this paper, we analyze the time-dependent extra spatial dimensions in six dimensional (6D) space-time. The 4-brane is assumed to be a de Sitter space. Based on the form of the brane-world energy-momentum tensor proposed by Shiromizu et al. and the five dimensions by Peter K. F. Kuhfittic, we extended the theory to the 2-codimension embedded in higher dimensions. The inflation scenario in 6D is investigated in two cases of cosmological constant: Ʌ > 0 and Ʌ < 0. The energy of two extra dimensions is calculated too.
A Four-Dimensional Continuum Theory of Space-Time and the Classical Physical Fields
Directory of Open Access Journals (Sweden)
Suhendro I.
2007-10-01
Full Text Available In this work, we attempt to describe the classical physical fields of gravity, electromagnetism, and the so-called intrinsic spin (chirality in terms of a set of fully geometrized constitutive equations. In our formalism, we treat the four-dimensional space-time continuum as a deformable medium and the classical fields as intrinsic stress and spin fields generated by infinitesimal displacements and rotations in the space-time continuum itself. In itself, the unifying continuum approach employed herein may suggest a possible unified field theory of the known classical physical fields.
Hesterberg, Stephen G; Duckett, C Cole; Salewski, Elizabeth A; Bell, Susan S
2017-01-31
Identifying and quantifying the relevant properties of habitat structure that mediate predator-prey interactions remains a persistent challenge. Most previous studies investigate effects of structural density on trophic interactions and typically quantify refuge quality using one or two-dimensional metrics. Few consider spatial arrangement of components (i.e., orientation and shape) and often neglect to measure the total three-dimensional (3D) space available as refuge. This study tests whether the three-dimensionality of interstitial space, an attribute produced by the spatial arrangement of oyster (Crassostrea virginica) shells, impacts the foraging success of nektonic predators (primary blue crab, Callinectes sapidus) on mud crab prey (Eurypanopeus depressus) in field and mesocosm experiments. Interstices of 3D-printed shell mimics were manipulated by changing either their orientation (angle) or internal shape (crevice or channel). In both field and mesocosm experiments, under conditions of constant structural density, predator foraging success was influenced by 3D aspects of interstitial space. Proportional survivorship of tethered mud crabs differed significantly as 3D interstitial space varied by orientation, displaying decreasing prey survivorship as angle of orientation increased (0° = 0.76, 22.5° = 0.13, 45° = 0.0). Tethered prey survivorship was high when 3D interstitial space of mimics was modified by internal shape (crevice = 0.89, channel = 0.96) and these values did not differ significantly. In mesocosms, foraging success of blue crabs varied with 3D interstitial space as mean proportional survivorship (± SE) of mud crabs was significantly lower in 45° (0.27 ± 0.06) versus 0° (0.86 ± 0.04) orientations and for crevice (0.52 ± 0.11) versus channel shapes (0.95 ± 0.01). These results suggest that 3D aspects of interstitial space, which have direct relevance to refuge quality, can strongly influence foraging success in our oyster reef habitat
Falomir, H.; Pisani, P. A. G.; Vega, F.; Cárcamo, D.; Méndez, F.; Loewe, M.
2016-02-01
We study two-dimensional Hamiltonians in phase space with noncommutativity both in coordinates and momenta. We consider the generator of rotations on the noncommutative plane and the Lie algebra generated by Hermitian rotationally invariant quadratic forms of noncommutative dynamical variables. We show that two quantum phases are possible, characterized by the Lie algebras {sl}(2,{{R}}) or su(2) according to the relation between the noncommutativity parameters, with the rotation generator related with the Casimir operator. From this algebraic perspective, we analyze the spectrum of some simple models with nonrelativistic rotationally invariant Hamiltonians in this noncommutative phase space, such as the isotropic harmonic oscillator, the Landau problem and the cylindrical well potential.
Falomir, H; Vega, F; Cárcamo, D; Méndez, F; Loewe, M
2015-01-01
We study two-dimensional Hamiltonians in phase space with noncommutativity both in coordinates and momenta. We consider the generator of rotations on the noncommutative plane and the Lie algebra generated by Hermitian rotationally invariant quadratic forms of noncommutative dynamical variables. We show that two quantum phases are possible, characterized by the Lie algebras $sl(2,\\mathbb{R})$ or $su(2)$ according to the relation between the noncommutativity parameters. From this perspective, we analyze the spectrum of some simple models with nonrelativistic rotationally invariant Hamiltonians in this noncommutative phase space, as the isotropic harmonic oscillator, the Landau problem and the cylindrical well potential.
Bolokhov, A A; Bolokhov, T A; Sherman, S G
1996-01-01
We present the analysis of the phase space geometry of 2 \\rightarrow 3 reaction for the general case of nonzero and unequal particle masses. Its purpose is to elaborate an alternative approach to the problem of integration over phase space which does not exploit the Monte Carlo principle. The fast and effective algorithm of integration based on Gauss method is developed for treating 1--dimensional distributions in two--particle invariant variables. The algorithm is characterized by significantly improved accuracy and it can meet requirements of interactive processing.
A Two-Dimensional Signal Space for Intensity-Modulated Channels
Karout, Johnny; Kschischang, Frank R; Agrell, Erik
2012-01-01
A two-dimensional signal space for intensity- modulated channels is presented. Modulation formats using this signal space are designed to maximize the minimum distance between signal points while satisfying average and peak power constraints. The uncoded, high-signal-to-noise ratio, power and spectral efficiencies are compared to those of the best known formats. The new formats are simpler than existing subcarrier formats, and are superior if the bandwidth is measured as 90% in-band power. Existing subcarrier formats are better if the bandwidth is measured as 99% in-band power.
Quantum homogeneous spaces and special functions with a dimensional deformation parameter
Bonechi, F.; Giachetti, R.; del Olmo, M. A.; Sorace, E.; Tarlini, M.
1996-12-01
We study the most elementary aspects of harmonic analysis on a homogeneous space of a deformation of the two-dimensional Euclidean group, admitting generalizations to dimensions three and four, whose quantum parameter has the physical dimensions of length. The homogeneous space is recognized as a new quantum plane and the action of the Euclidean quantum group is used to determine an eigenvalue problem for the Casimir operator, which constitutes the analogue of the Schrödinger equation in the presence of such a deformation. The solutions are given in the plane-wave and angular-momentum bases and are expressed in terms of hypergeometric series with non-commuting parameters.
Multilayer adsorption on fractal surfaces.
Vajda, Péter; Felinger, Attila
2014-01-10
Multilayer adsorption is often observed in liquid chromatography. The most frequently employed model for multilayer adsorption is the BET isotherm equation. In this study we introduce an interpretation of multilayer adsorption measured on liquid chromatographic stationary phases based on the fractal theory. The fractal BET isotherm model was successfully used to determine the apparent fractal dimension of the adsorbent surface. The nonlinear fitting of the fractal BET equation gives us the estimation of the adsorption equilibrium constants and the monolayer saturation capacity of the adsorbent as well. In our experiments, aniline and proline were used as test molecules on reversed phase and normal phase columns, respectively. Our results suggest an apparent fractal dimension 2.88-2.99 in the case of reversed phase adsorbents, in the contrast with a bare silica column with a fractal dimension of 2.54.
Kinetic properties of fractal media
Chumak, Oleg V
2016-01-01
Kinetic processes in fractal stellar media are analyzed in terms of the approach developed in our earlier paper (Chumak, Rastorguev, 2016) involving a generalization of the nearest neighbor and random force distributions to fractal media. Diffusion is investigated in the approximation of scale-dependent conditional density based on an analysis of the solutions of the corresponding Langevin equations. It is shown that kinetic parameters (time scales, coefficients of dynamic friction, diffusion, etc.) for fractal stellar media can differ significantly both qualitatively and quantitatively from the corresponding parameters for a quasi-uniform random media with limited fluctuations. The most important difference is that in the fractal case kinetic parameters depend on spatial scale length and fractal dimension of the medium studied. A generalized kinetic equation for stellar media (fundamental equation of stellar dynamics) is derived in the Fokker-Planck approximation with the allowance for the fractal properties...
Fractals in geology and geophysics
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.
Fractals in several electrode materials
Energy Technology Data Exchange (ETDEWEB)
Zhang, Chunyong, E-mail: zhangchy@njau.edu.cn [Department of Chemistry, College of Science, Nanjing Agricultural University, Nanjing 210095 (China); Suzhou Key Laboratory of Environment and Biosafety, Suzhou Academy of Southeast University, Dushuhu lake higher education town, Suzhou 215123 (China); Wu, Jingyu [Department of Chemistry, College of Science, Nanjing Agricultural University, Nanjing 210095 (China); Fu, Degang [Suzhou Key Laboratory of Environment and Biosafety, Suzhou Academy of Southeast University, Dushuhu lake higher education town, Suzhou 215123 (China); State Key Laboratory of Bioelectronics, Southeast University, Nanjing 210096 (China)
2014-09-15
Highlights: • Fractal geometry was employed to characterize three important electrode materials. • The surfaces of all studied electrodes were proved to be very rough. • The fractal dimensions of BDD and ACF were scale dependent. • MMO film was more uniform than BDD and ACF in terms of fractal structures. - Abstract: In the present paper, the fractal properties of boron-doped diamond (BDD), mixed metal oxide (MMO) and activated carbon fiber (ACF) electrode have been studied by SEM imaging at different scales. Three materials are self-similar with mean fractal dimension in the range of 2.6–2.8, confirming that they all exhibit very rough surfaces. Specifically, it is found that MMO film is more uniform in terms of fractal structure than BDD and ACF. As a result, the intriguing characteristics make these electrodes as ideal candidates for high-performance decontamination processes.
Equivalence classes of the 3rd Grassman space over a 5-dimensional vector space
Directory of Open Access Journals (Sweden)
Kuldip Singh
1978-01-01
Full Text Available An equivalence relation is defined on ΛrV, the rth Grassman space over V and the problem of the determnation of the equivalence classes defined by this relation is considered. For any r and V, the decomposable elements form an equivalence class. For r=2, the length of the element determines the equivalence class that it is in. Elements of the same length are equivalent, those of unequal lengths are inequivalent. When r≥3, the length is no longer a sufficient indicator, except when the length is one. Besides these general questions, the equivalence classes of Λ3V, when dimV=5 are determined.
Reconstruction of gyrotropic phase-space distributions from one-dimensional projections
DEFF Research Database (Denmark)
Egedal, J.; Bindslev, H.
2004-01-01
This paper describes mathematical tools applicable to the reconstruction of anisotropic velocity distributions through the unfolding of data coming from techniques like collective Thomson scattering or laser induced fluorescence, where one-dimensional projections of the velocity space along....... An example is given based on alpha particle distribution in the Joint European Torus tokamak [P. H. Rebut and B. E. Keen, Fusion Technol. 11, 13 (1987)]. (C) 2004 American Institute of Physics....
Exact solutions of the (3+1)-dimensional space-time fractional Jimbo-Miwa equation
Aksoy, Esin; Guner, Ozkan; Bekir, Ahmet; Cevikel, Adem C.
2016-06-01
Exact solutions of the (3+1)-dimensional space-time fractional Jimbo-Miwa equation are studied by the generalized Kudryashov method, the exp-function method and the (G'/G)-expansion method. The solutions obtained include the form of hyperbolic functions, trigonometric and rational functions. These methods are effective, simple, and many types of solutions can be obtained at the same time.
Integration over an Infinite-Dimensional Banach Space and Probabilistic Applications
Directory of Open Access Journals (Sweden)
Claudio Asci
2014-01-01
Full Text Available We study, for some subsets I of N*, the Banach space E of bounded real sequences {xn}n∈I. For any integer k, we introduce a measure over (E,B(E that generalizes the k-dimensional Lebesgue measure; consequently, also a theory of integration is defined. The main result of our paper is a change of variables' formula for the integration.
Phase Space Compression in One-Dimensional Complex Ginzburg-Landau Dquation
Institute of Scientific and Technical Information of China (English)
GAO Ji-Hua; PENG Jian-Hua
2007-01-01
The transition from stationary to oscillatory states in dynamical systems under phase space compression is investigated. By considering the model for the spatially one-dimensional complex Ginzburg-Landau equation, we find that defect turbulence can be substituted with stationary and oscillatory signals by applying system perturbation and confining variable into various ranges. The transition procedure described by the oscillatory frequency is studied via numerical simulations in detail.
A framework for analyzing ecological trait-based models in multi-dimensional niche spaces
Biancalani, Tommaso; DeVille, Lee; Goldenfeld, Nigel
2014-01-01
We develop an theoretical approach for predicting biodiversity in multi-dimensional niche spaces, arising due to ecological drivers such as competitive exclusion. The novelty of our approach relies on the fact that ecological niches are described by sequences of strings, which allows us to describe multiple traits. We define the mathematical framework for analyzing pattern forming instabilities in these models, showing surprisingly that the analytic linear theory predicts the asymptotically l...
Institute of Scientific and Technical Information of China (English)
Lou Zhi-Mei; Chen Zi-Dong; Wang Wen-Long
2005-01-01
In this paper, we express the differential equations of a noncentral dynamical system in Ermakov formalism to obtain the Ermakov invariant. In term of Hamiltonian theories and using the Ermakov invariant as the Hamiltonian,the Poisson structure of a noncentral dynamical system in four-dimensional phase space are constructed. The result indicates that the Poisson structure is degenerate and the noncentral dynamical system possesses four invariants: the Hamiltonian, the Ermakov invariant and two Casimir functions.
Free massless fermionic fields of arbitrary spin in d-dimensional anti-de Sitter space
Energy Technology Data Exchange (ETDEWEB)
Vasiliev, M.A.
1988-04-25
Free massless fermionic fields of arbitrary spins, corresponding to fully symmetric tensor-spinor irreducible representations of the flat little group SO(d-2), are described in d-dimensional anti-de Sitter space in terms of differential forms. Appropriate linearized higher-spin curvature 2-forms are found. Explicitly gauge invariant higher-spin actions are constructed in terms of these linearized curvatures.
Institute of Scientific and Technical Information of China (English)
Tian Hongwei; Jing Zhongliang; Hu Shiqiang; Li Jianxun
2005-01-01
A new fusion tracking algorithm is presented to track maneuvering target in three-dimensional (3D) space with bearings-only measurements. With the introduction of passive location and interacting multiple model (IMM) algorithm based on multirate model, the high-rate sequence measurements of two sensors are utilized. Simulation results show that the performance of tracking has been improved. The new algorithm removes the barrier of processing high-rate bearingsonly measurements.
Numerical Relativity in D dimensional space-times: Collisions of unequal mass black holes
Energy Technology Data Exchange (ETDEWEB)
Witek, Helvi; Cardoso, Vitor; Sperhake, Ulrich [CENTRA, Departamento de Fisica, Instituto Superior Tecnico, Universidade Tecnica de Lisboa - UTL, Av. Rovisco Pais 1, 1049 Lisboa (Portugal); Gualtieri, Leonardo [Dipartimento di Fisica, Universita di Roma ' Sapienza' and Sezione INFN Roma1, P.A. Moro 5, 00185, Roma (Italy); Herdeiro, Carlos [Departamento de Fisica da Universidade de Aveiro, Campus de Santiago, 3810-183 Aveiro (Portugal); Zilhao, Miguel, E-mail: helvi.witek@ist.utl.pt [Centro de Fisica do Porto - CFP, Departamento de Fisica e Astronomia, Faculdade de Ciencias da Universidade do Porto - FCUP, Rua do Campo Alegre, 4169-007 Porto (Portugal)
2011-09-22
We present unequal mass head-on collisions of black holes in D = 5 dimensional space-times. We have simulated BH systems with mass ratios q 1,1/2, 1/3, 1/4. We extract the total energy radiated throughout the collision and compute the linear momentum flux and the recoil velocity of the final black hole. The numerical results show very good agreement with point particle calculations when extrapolated to this limit.
Keller, Kai Johannes
2010-01-01
The present work contains a consistent formulation of the methods of dimensional regularization (DimReg) and minimal subtraction (MS) in Minkowski position space. The methods are implemented into the framework of perturbative Algebraic Quantum Field Theory (pAQFT). The developed methods are used to solve the Epstein-Glaser recursion for the construction of time-ordered products in all orders of causal perturbation theory. A solution is given in terms of a forest formula in the sense of Zimmer...
Dütsch, Michael; Fredenhagen, Klaus; Keller, Kai J.; Rejzner, Katarzyna Anna
2013-01-01
We reformulate dimensional regularization as a regularization method in position space and show that it can be used to give a closed expression for the renormalized time-ordered products as solutions to the induction scheme of Epstein-Glaser. For scalar fields the resulting renormalization method is always applicable, we compute several examples. We also analyze the Hopf algebraic aspects of the combinatorics. Our starting point is the Main Theorem of Renormalization of Stora and Popineau and...
Fast-forward scaling in a finite-dimensional Hilbert space
TAKAHASHI, Kazutaka
2014-01-01
Time evolution of quantum systems is accelerated by the fast-forward scaling. We reformulate the method to study systems in a finite-dimensional Hilbert space. For several simple systems, we explicitly construct the acceleration potential. We also use our formulation to accelerate the adiabatic dynamics. Applying the method to the transitionless quantum driving, we find that the fast-forward potential can be understood as a counterdiabatic term.
Dynamics of a harmonic oscillator in a finite-dimensional Hilbert space
Energy Technology Data Exchange (ETDEWEB)
Kuang Leman (CCAST (World Lab.), Beijing, BJ (China) Dept. of Physics and Inst. of Physics, Hunan Normal Univ. (China)); Wang Fabo (Dept. of Physics, Hunan Normal Univ. (China)); Zhou Yanguo (Dept. of Physics, Hunan Normal Univ. (China))
1993-11-29
Some dynamical properties of a finite-dimensional Hilbert space harmonic oscillator (FDHSHO) are studied. The time evolution of the position and momentum operators and the second-order quadrature squeezing are investigated in detail. It is shown that the coherent states of the FDHSHO are not the minimum uncertainty states of the position and momentum operators of the FDHSHO. It is found that the second-order squeezing of the quadrature operators vanishes and reappears periodically in the time evolution. (orig.)
Subjective refraction techniques in the frame of the three-dimensional dioptric space.
Muñoz-Escrivá, L; Furlan, W D
2001-02-01
A novel heuristic approach to the well-known representation of the dioptric power in a three-dimensional space is presented. It is shown how this theoretical framework is ideal for discussing the principles of several subjective refraction methods. In particular, this formalism is used to justify the stenopaic slit refraction, the Barnes subjective refraction technique, and the Jackson cross-cylinder procedure. In view of this analysis, some modifications to the traditional procedures are proposed.
Lee, Jenny Hyunjung; McDonnell, Kevin T; Zelenyuk, Alla; Imre, Dan; Mueller, Klaus
2014-03-01
Although the euclidean distance does well in measuring data distances within high-dimensional clusters, it does poorly when it comes to gauging intercluster distances. This significantly impacts the quality of global, low-dimensional space embedding procedures such as the popular multidimensional scaling (MDS) where one can often observe nonintuitive layouts. We were inspired by the perceptual processes evoked in the method of parallel coordinates which enables users to visually aggregate the data by the patterns the polylines exhibit across the dimension axes. We call the path of such a polyline its structure and suggest a metric that captures this structure directly in high-dimensional space. This allows us to better gauge the distances of spatially distant data constellations and so achieve data aggregations in MDS plots that are more cognizant of existing high-dimensional structure similarities. Our biscale framework distinguishes far-distances from near-distances. The coarser scale uses the structural similarity metric to separate data aggregates obtained by prior classification or clustering, while the finer scale employs the appropriate euclidean distance.
Efficient and accurate nearest neighbor and closest pair search in high-dimensional space
Tao, Yufei
2010-07-01
Nearest Neighbor (NN) search in high-dimensional space is an important problem in many applications. From the database perspective, a good solution needs to have two properties: (i) it can be easily incorporated in a relational database, and (ii) its query cost should increase sublinearly with the dataset size, regardless of the data and query distributions. Locality-Sensitive Hashing (LSH) is a well-known methodology fulfilling both requirements, but its current implementations either incur expensive space and query cost, or abandon its theoretical guarantee on the quality of query results. Motivated by this, we improve LSH by proposing an access method called the Locality-Sensitive B-tree (LSB-tree) to enable fast, accurate, high-dimensional NN search in relational databases. The combination of several LSB-trees forms a LSB-forest that has strong quality guarantees, but improves dramatically the efficiency of the previous LSH implementation having the same guarantees. In practice, the LSB-tree itself is also an effective index which consumes linear space, supports efficient updates, and provides accurate query results. In our experiments, the LSB-tree was faster than: (i) iDistance (a famous technique for exact NN search) by two orders ofmagnitude, and (ii) MedRank (a recent approximate method with nontrivial quality guarantees) by one order of magnitude, and meanwhile returned much better results. As a second step, we extend our LSB technique to solve another classic problem, called Closest Pair (CP) search, in high-dimensional space. The long-term challenge for this problem has been to achieve subquadratic running time at very high dimensionalities, which fails most of the existing solutions. We show that, using a LSB-forest, CP search can be accomplished in (worst-case) time significantly lower than the quadratic complexity, yet still ensuring very good quality. In practice, accurate answers can be found using just two LSB-trees, thus giving a substantial
Turbulent wakes of fractal objects.
Staicu, Adrian; Mazzi, Biagio; Vassilicos, J C; van de Water, Willem
2003-06-01
Turbulence of a windtunnel flow is stirred using objects that have a fractal structure. The strong turbulent wakes resulting from three such objects which have different fractal dimensions are probed using multiprobe hot-wire anemometry in various configurations. Statistical turbulent quantities are studied within inertial and dissipative range scales in an attempt to relate changes in their self-similar behavior to the scaling of the fractal objects.
Partially ordered sets, transfinite topology and the dimension of Cantorian-fractal spacetime
Energy Technology Data Exchange (ETDEWEB)
Marek-Crnjac, L. [Institute of Mathematics and Physics, University of Maribor (Slovenia)], E-mail: leila.marek@guest.arnes.si
2009-11-15
We introduce partially ordered sets and relate them to random Cantor sets of E-infinity theory. Subsequently we derive the dimensionality of Cantorian-fractal spacetime using posets and E-infinity transfinite Cantor sets.
Wang, Bin; Shi, Wenzhong; Miao, Zelang
2015-01-01
Standard deviational ellipse (SDE) has long served as a versatile GIS tool for delineating the geographic distribution of concerned features. This paper firstly summarizes two existing models of calculating SDE, and then proposes a novel approach to constructing the same SDE based on spectral decomposition of the sample covariance, by which the SDE concept is naturally generalized into higher dimensional Euclidean space, named standard deviational hyper-ellipsoid (SDHE). Then, rigorous recursion formulas are derived for calculating the confidence levels of scaled SDHE with arbitrary magnification ratios in any dimensional space. Besides, an inexact-newton method based iterative algorithm is also proposed for solving the corresponding magnification ratio of a scaled SDHE when the confidence probability and space dimensionality are pre-specified. These results provide an efficient manner to supersede the traditional table lookup of tabulated chi-square distribution. Finally, synthetic data is employed to generate the 1-3 multiple SDEs and SDHEs. And exploratory analysis by means of SDEs and SDHEs are also conducted for measuring the spread concentrations of Hong Kong's H1N1 in 2009.
Directory of Open Access Journals (Sweden)
Bin Wang
Full Text Available Standard deviational ellipse (SDE has long served as a versatile GIS tool for delineating the geographic distribution of concerned features. This paper firstly summarizes two existing models of calculating SDE, and then proposes a novel approach to constructing the same SDE based on spectral decomposition of the sample covariance, by which the SDE concept is naturally generalized into higher dimensional Euclidean space, named standard deviational hyper-ellipsoid (SDHE. Then, rigorous recursion formulas are derived for calculating the confidence levels of scaled SDHE with arbitrary magnification ratios in any dimensional space. Besides, an inexact-newton method based iterative algorithm is also proposed for solving the corresponding magnification ratio of a scaled SDHE when the confidence probability and space dimensionality are pre-specified. These results provide an efficient manner to supersede the traditional table lookup of tabulated chi-square distribution. Finally, synthetic data is employed to generate the 1-3 multiple SDEs and SDHEs. And exploratory analysis by means of SDEs and SDHEs are also conducted for measuring the spread concentrations of Hong Kong's H1N1 in 2009.
Spinor formalism and the geometry of six-dimensional Riemannian spaces
Andreev, K V
2012-01-01
The article consists of the Russian and English variants of Ph.D. Thesis in which the answers is given on the following questions: 1. how to construct the spinor formalism for n=6; 2. how to construct the spinor formalism for n=8; 3. how to prolong the Riemannian connection from the tangent bundle into the spinor one with the base: a complex analytical 6-dimensional Riemannian space; 4. how to construct the real and complex representations of this bundles; 5. how to construct the curvature spinors and to investigate its properties; 6. how to obtain the canonical form of a bilinear form for the 6-dimensional pseudo-Euclidean space with the even index of the metric; 7. how to construct the geometric interpretation of isotropic twistors on the isotropic cone of the 6-dimensional pseudo-Euclidean space with the index equal to 4; 8. how to construct the generalization of the Cartan triality principle to the Klein correspondence; 9. how to construct the structural constants of the octonion algebra for the initial i...
Exploring Replica-Exchange Wang-Landau sampling in higher-dimensional parameter space
Valentim, Alexandra; Rocha, Julio C. S.; Tsai, Shan-Ho; Li, Ying Wai; Eisenbach, Markus; Fiore, Carlos E.; Landau, David P.
2015-09-01
We considered a higher-dimensional extension for the replica-exchange Wang- Landau algorithm to perform a random walk in the energy and magnetization space of the two-dimensional Ising model. This hybrid scheme combines the advantages of Wang-Landau and Replica-Exchange algorithms, and the one-dimensional version of this approach has been shown to be very efficient and to scale well, up to several thousands of computing cores. This approach allows us to split the parameter space of the system to be simulated into several pieces and still perform a random walk over the entire parameter range, ensuring the ergodicity of the simulation. Previous work, in which a similar scheme of parallel simulation was implemented without using replica exchange and with a different way to combine the result from the pieces, led to discontinuities in the final density of states over the entire range of parameters. From our simulations, it appears that the replica-exchange Wang-Landau algorithm is able to overcome this difficulty, allowing exploration of higher parameter phase space by keeping track of the joint density of states.
Exploring Replica-Exchange Wang-Landau sampling in higher-dimensional parameter space
Energy Technology Data Exchange (ETDEWEB)
Valentim, Alexandra [University of Georgia, Athens, GA; Rocha, Julio C. S. [Universidade Federal de Minas Gerais; Tsai, Shan-Ho [University of Georgia, Athens, GA; Li, Ying Wai [ORNL; Eisenbach, Markus [ORNL; Fiore, Carlos E [University of Sao Paulo, BRAZIL; Landau, David P [University of Georgia, Athens, GA
2015-01-01
We considered a higher-dimensional extension for the replica-exchange Wang-Landau algorithm to perform a random walk in the energy and magnetization space of the two-dimensional Ising model. This hybrid scheme combines the advantages of Wang-Landau and Replica-Exchange algorithms, and the one-dimensional version of this approach has been shown to be very efficient and to scale well, up to several thousands of computing cores. This approach allows us to split the parameter space of the system to be simulated into several pieces and still perform a random walk over the entire parameter range, ensuring the ergodicity of the simulation. Previous work, in which a similar scheme of parallel simulation was implemented without using replica exchange and with a different way to combine the result from the pieces, led to discontinuities in the final density of states over the entire range of parameters. From our simulations, it appears that the replica-exchange Wang-Landau algorithm is able to overcome this diculty, allowing exploration of higher parameter phase space by keeping track of the joint density of states.
Statistical mechanics and fractals
Dobrushin, Roland Lvovich
1993-01-01
This book is composed of two texts, by R.L. Dobrushin and S. Kusuoka, each representing the content of a course of lectures given by the authors. They are pitched at graduate student level and are thus very accessible introductions to their respective subjects for students and non specialists. CONTENTS: R.L. Dobrushin: On the Way to the Mathematical Foundations of Statistical Mechanics.- S. Kusuoka: Diffusion Processes on Nested Fractals.
Martin, Demetri
2015-03-01
Demetri Maritn prepared this palindromic poem as his project for Michael Frame's fractal geometry class at Yale. Notice the first, fourth, and seventh words in the second and next-to-second lines are palindromes, the first two and last two lines are palindromes, the middle line, "Be still if I fill its ebb" minus its last letter is a palindrome, and the entire poem is a palindrome...
Fractal multifiber microchannel plates
Cook, Lee M.; Feller, W. B.; Kenter, Almus T.; Chappell, Jon H.
1992-01-01
The construction and performance of microchannel plates (MCPs) made using fractal tiling mehtods are reviewed. MCPs with 40 mm active areas having near-perfect channel ordering were produced. These plates demonstrated electrical performance characteristics equivalent to conventionally constructed MCPs. These apparently are the first MCPs which have a sufficiently high degree of order to permit single channel addressability. Potential applications for these devices and the prospects for further development are discussed.
Fractal Risk Assessment of ISS Propulsion Module in Meteoroid and Orbital Debris Environments
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.
Spatial log-periodic oscillations of first-passage observables in fractals
Akkermans, Eric; Benichou, Olivier; Dunne, Gerald V.; Teplyaev, Alexander; Voituriez, Raphael
2012-12-01
For transport processes in geometrically restricted domains, the mean first-passage time (MFPT) admits a general scaling dependence on space parameters for diffusion, anomalous diffusion, and diffusion in disordered or fractal media. For transport in self-similar fractal structures, we obtain an expression for the source-target distance dependence of the MFPT that exhibits both the leading power-law behavior, depending on the Hausdorff and spectral dimension of the fractal, as well as small log-periodic oscillations that are a clear and definitive signal of the underlying fractal structure. We also present refined numerical results for the Sierpinski gasket that confirm this oscillatory behavior.
Darwinian Evolution and Fractals
Carr, Paul H.
2009-05-01
Did nature's beauty emerge by chance or was it intelligently designed? Richard Dawkins asserts that evolution is blind aimless chance. Michael Behe believes, on the contrary, that the first cell was intelligently designed. The scientific evidence is that nature's creativity arises from the interplay between chance AND design (laws). Darwin's ``Origin of the Species,'' published 150 years ago in 1859, characterized evolution as the interplay between variations (symbolized by dice) and the natural selection law (design). This is evident in recent discoveries in DNA, Madelbrot's Fractal Geometry of Nature, and the success of the genetic design algorithm. Algorithms for generating fractals have the same interplay between randomness and law as evolution. Fractal statistics, which are not completely random, characterize such phenomena such as fluctuations in the stock market, the Nile River, rainfall, and tree rings. As chaos theorist Joseph Ford put it: God plays dice, but the dice are loaded. Thus Darwin, in discovering the evolutionary interplay between variations and natural selection, was throwing God's dice!
Spina, Maria E; Saraceno, Marcos
2010-01-01
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.
Fractals a very short introduction
Falconer, Kenneth
2013-01-01
Many are familiar with the beauty and ubiquity of fractal forms within nature. Unlike the study of smooth forms such as spheres, fractal geometry describes more familiar shapes and patterns, such as the complex contours of coastlines, the outlines of clouds, and the branching of trees. In this Very Short Introduction, Kenneth Falconer looks at the roots of the 'fractal revolution' that occurred in mathematics in the 20th century, presents the 'new geometry' of fractals, explains the basic concepts, and explores the wide range of applications in science, and in aspects of economics.This is esse
Relativistic star solutions in higher-dimensional pseudospheroidal space-time
Indian Academy of Sciences (India)
P K Chattopadhyay; B C Paul
2010-04-01
We obtain relativistic solutions of a class of compact stars in hydrostatic equilibrium in higher dimensions by assuming a pseudospheroidal geometry for the space-time. The space-time geometry is assumed to be ( - 1) pseudospheroid immersed in a -dimensional Euclidean space. The spheroidicity parameter () plays an important role in determining the equation of state of the matter content and the maximum radius of such stars. It is found that the core density of compact objects is approximately proportional to the square of the space-time dimensions (), i.e., core of the star is denser in higher dimensions than that in conventional four dimensions. The central density of a compact star is also found to depend on the parameter . One obtains a physically interesting solution satisfying the acoustic condition when lies in the range > ( + 1)/( − 3) for the space-time dimensions ranging from = 4 to 8 and ( + 1)/( − 3) < < (2 - 4 + 3)/(2 - 8 - 1) for space-time dimensions ≥ 9. The non-negativity of the energy density () constrains the parameter with a lower limit (> 1). We note that in the case of a superdense compact object the number of space-time dimensions cannot be taken infinitely large, which is a different result from the braneworld model.
Fractal properties of the lattice Lotka-Volterra model.
Tsekouras, G A; Provata, A
2002-01-01
The lattice Lotka-Volterra (LLV) model is studied using mean-field analysis and Monte Carlo simulations. While the mean-field phase portrait consists of a center surrounded by an infinity of closed trajectories, when the process is restricted to a two-dimensional (2D) square lattice, local inhomogeneities/fluctuations appear. Spontaneous local clustering is observed on lattice and homogeneous initial distributions turn into clustered structures. Reactions take place only at the interfaces between different species and the borders adopt locally fractal structure. Intercluster surface reactions are responsible for the formation of local fluctuations of the species concentrations. The box-counting fractal dimension of the LLV dynamics on a 2D support is found to depend on the reaction constants while the upper bound of fractality determines the size of the local oscillators. Lacunarity analysis is used to determine the degree of clustering of homologous species. Besides the spontaneous clustering that takes place on a regular 2D lattice, the effects of fractal supports on the dynamics of the LLV are studied. For supports of dimensionality D(s)<2 the lattice can, for certain domains of the reaction constants, adopt a poisoned state where only one of the species survives. By appropriately selecting the fractal dimension of the substrate, it is possible to direct the system into a poisoned or oscillatory steady state at will.
Institute of Scientific and Technical Information of China (English)
Ran SHEN; Yu Cai SU
2007-01-01
We show that the support of an irreducible weight module over the twisted Heisenberg-Virasoro algebra, which has an infinite-dimensional weight space, coincides with the weight lattice and that all nontrivial weight spaces of such a module are infinite dimensional. As a corollary, we obtain that every irreducible weight module over the twisted Heisenberg-Virasoro algebra, having a nontrivial finite-dimensional weight space, is a Harish-Chandra module (and hence is either an irreducible highest or lowest weight module or an irreducible module from the intermediate series).
Euclidean and fractal geometry of microvascular networks in normal and neoplastic pituitary tissue.
Di Ieva, Antonio; Grizzi, Fabio; Gaetani, Paolo; Goglia, Umberto; Tschabitscher, Manfred; Mortini, Pietro; Rodriguez y Baena, Riccardo
2008-07-01
In geometrical terms, tumour vascularity is an exemplary anatomical system that irregularly fills a three-dimensional Euclidean space. This physical characteristic and the highly variable shapes of the vessels lead to considerable spatial and temporal heterogeneity in the delivery of oxygen, nutrients and drugs, and the removal of metabolites. Although these biological characteristics are well known, quantitative analyses of newly formed vessels in two-dimensional histological sections still fail to view their architecture as a non-Euclidean geometrical entity, thus leading to errors in visual interpretation and discordant results from different laboratories concerning the same tumour. We here review the literature concerning microvessel density estimates (a Euclidean-based approach quantifying vascularity in normal and neoplastic pituitary tissues) and compare the results. We also discuss the limitations of Euclidean quantitative analyses of vascularity and the helpfulness of a fractal geometry-based approach as a better means of quantifying normal and neoplastic pituitary microvasculature.
Thermodynamic fractals and formalism. Fractales y formalismo termodinamico
Energy Technology Data Exchange (ETDEWEB)
Chacon, R.; Morales, J.J.
1994-01-01
We give a brief introduction to the so called ''thermodynamical description of fractals'' restricting our attention to Cantor sets generated by chaotic motion of a dynamical system. In particular, an entropy function and a free energy are introduced for multi fractals. (Author) 14 refs.
An Eternal Time Machine in 2+1 Dimensional anti-de Sitter Space
De Deo, S
2002-01-01
2+1 dimensional anti-de Sitter space has been the subject of much recent investigation. Studies of the behaviour of point particles in this space have given us a greater understanding of the BTZ black hole solutions produced by topological identification of adS isometries. In this paper, we present a new configuration of two orbiting massive point particles that leads to an ``eternal'' time machine, where closed timelike curves fill the entire space. In contrast to previous solutions, this configuration has no event or chronology horizons. Another interesting feature is that there is no lower bound on the relative velocities of the point masses used to construct the time machine; as long as the particles exceed a certain mass threshold, an eternal time machine will be produced.
Fractality and geometry in ultra-relativistic nuclear collisions
Zborovský, I
2002-01-01
Assuming fractality of hadronic constituents, we argue that asymmetry of space-time can be induced in the ultra-relativistic interactions of hadrons and nuclei. The asymmetry is expressed in terms of the anomalous fractal dimensions of the colliding objects. Besides state of motion, the relativistic principle is applied to the state of asymmetry as well. Such realization of relativity concerns scale dependence of physical laws emerging at small distances. We show that induced asymmetries of space-time are a priori not excluded by the Michelson's experiment even at large scales.
Possibilities of identifying cyber attack in noisy space of n-dimensional abstract system
Energy Technology Data Exchange (ETDEWEB)
Jašek, Roman; Dvořák, Jiří; Janková, Martina; Sedláček, Michal [Tomas Bata University in Zlin Nad Stranemi 4511, 760 05 Zlin, Czech republic jasek@fai.utb.cz, dvorakj@aconte.cz, martina.jankova@email.cz, michal.sedlacek@email.cz (Czech Republic)
2016-06-08
This article briefly mentions some selected options of current concept for identifying cyber attacks from the perspective of the new cyberspace of real system. In the cyberspace, there is defined n-dimensional abstract system containing elements of the spatial arrangement of partial system elements such as micro-environment of cyber systems surrounded by other suitably arranged corresponding noise space. This space is also gradually supplemented by a new image of dynamic processes in a discreet environment, and corresponding again to n-dimensional expression of time space defining existence and also the prediction for expected cyber attacksin the noise space. Noises are seen here as useful and necessary for modern information and communication technologies (e.g. in processes of applied cryptography in ICT) and then the so-called useless noises designed for initial (necessary) filtering of this highly aggressive environment and in future expectedly offensive background in cyber war (e.g. the destruction of unmanned means of an electromagnetic pulse, or for destruction of new safety barriers created on principles of electrostatic field or on other principles of modern physics, etc.). The key to these new options is the expression of abstract systems based on the models of microelements of cyber systems and their hierarchical concept in structure of n-dimensional system in given cyberspace. The aim of this article is to highlight the possible systemic expression of cyberspace of abstract system and possible identification in time-spatial expression of real environment (on microelements of cyber systems and their surroundings with noise characteristics and time dimension in dynamic of microelements’ own time and externaltime defined by real environment). The article was based on a partial task of faculty specific research.
Possibilities of identifying cyber attack in noisy space of n-dimensional abstract system
Jašek, Roman; Dvořák, Jiří; Janková, Martina; Sedláček, Michal
2016-06-01
This article briefly mentions some selected options of current concept for identifying cyber attacks from the perspective of the new cyberspace of real system. In the cyberspace, there is defined n-dimensional abstract system containing elements of the spatial arrangement of partial system elements such as micro-environment of cyber systems surrounded by other suitably arranged corresponding noise space. This space is also gradually supplemented by a new image of dynamic processes in a discreet environment, and corresponding again to n-dimensional expression of time space defining existence and also the prediction for expected cyber attacksin the noise space. Noises are seen here as useful and necessary for modern information and communication technologies (e.g. in processes of applied cryptography in ICT) and then the so-called useless noises designed for initial (necessary) filtering of this highly aggressive environment and in future expectedly offensive background in cyber war (e.g. the destruction of unmanned means of an electromagnetic pulse, or for destruction of new safety barriers created on principles of electrostatic field or on other principles of modern physics, etc.). The key to these new options is the expression of abstract systems based on the models of microelements of cyber systems and their hierarchical concept in structure of n-dimensional system in given cyberspace. The aim of this article is to highlight the possible systemic expression of cyberspace of abstract system and possible identification in time-spatial expression of real environment (on microelements of cyber systems and their surroundings with noise characteristics and time dimension in dynamic of microelements' own time and externaltime defined by real environment). The article was based on a partial task of faculty specific research.
Fractal analysis: methodologies for biomedical researchers.
Ristanović, Dusan; Milosević, Nebojsa T
2012-01-01
Fractal analysis has become a popular method in all branches of scientific investigations including biology and medicine. Although there is a growing interest in the application of fractal analysis in biological sciences, questions about the methodology of fractal analysis have partly restricted its wider and comprehensible application. It is a notable fact that fractal analysis is derived from fractal geometry, but there are some unresolved issues that need to be addressed. In this respect, we discuss several related underlying principles for fractal analysis and establish the meaningful relationship between fractal analysis and fractal geometry. Since some concepts in fractal analysis are determined descriptively and/or qualitatively, this paper provides their exact mathematical definitions or explanations. Another aim of this study is to show that nowadays fractal analysis is an independent mathematical and experimental method based on Mandelbrot's fractal geometry, Euclidean traditiontal geometry and Richardson's coastline method.
Directory of Open Access Journals (Sweden)
Cahill R. T.
2014-10-01
Full Text Available Space speed fluctuations, which have a 1 / f spectrum, are shown to be the cause of solar flares. The direction and magnitude of the space flow has been detected from numer- ous different experimental techniques, and is close to the normal to the plane of the ecliptic. Zener diode data shows that the fluctuations in the space speed closely match the Sun Solar Cycle 23 flare count, and reveal that major solar flares follow major space speed fluctuations by some 6 days. This implies that a warning period of some 5 days in predicting major solar flares is possible using such detectors. This has significant conse- quences in being able to protect various spacecraft and Earth located electrical systems from the subsequent arrival of ejected plasma from a solar flare. These space speed fluctuations are the actual gravitational waves, and have a significant magnitude. This discovery is a significant application of the dynamical space phenomenon and theory. We also show that space flow turbulence impacts on the Earth’s climate, as such tur- bulence can input energy into systems, which is the basis of the Zener Diode Quantum Detector. Large scale space fluctuations impact on both the sun and the Earth, and as well explain temperature correlations with solar activity, but that the Earth temperatures are not caused by such solar activity. This implies that the Earth climate debate has been missing a key physical process. Observed diminishing gravitational waves imply a cooling epoch for the Earth for the next 30 years.
Multispectral image fusion based on fractal features
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
Exploring Replica-Exchange Wang-Landau sampling in higher-dimensional parameter space
Valentim, Alexandra; Tsai, Shan-Ho; Li, Ying Wai; Eisenbach, Markus; Fiore, Carlos E; Landau, David P
2015-01-01
We considered a higher-dimensional extension for the replica-exchange Wang-Landau algorithm to perform a random walk in the energy and magnetization space of the two-dimensional Ising model. This hybrid scheme combines the advantages of Wang-Landau and Replica-Exchange algorithms, and the one-dimensional version of this approach has been shown to be very efficient and to scale well, up to several thousands of computing cores. This approach allows us to split the parameter space of the system to be simulated into several pieces and still perform a random walk over the entire parameter range, ensuring the ergodicity of the simulation. Previous work, in which a similar scheme of parallel simulation was implemented without using replica exchange and with a different way to combine the result from the pieces, led to discontinuities in the final density of states over the entire range of parameters. From our simulations, it appears that the replica-exchange Wang-Landau algorithm is able to overcome this difficulty,...
Quantum-optical states in finite-dimensional Hilbert space; 1, General formalism
Miranowicz, A; Imoto, N; Miranowicz, Adam; Leonski, Wieslaw; Imoto, Nobuyuki
2001-01-01
The interest in quantum-optical states confined in finite-dimensional Hilbert spaces has recently been stimulated by the progress in quantum computing, quantum-optical state preparation and measurement techniques, in particular, by the development of the discrete quantum-state tomography. In the first part of our review we present two essentially different approaches to define harmonic oscillator states in the finite-dimensional Hilbert spaces. One of them is related to the truncation scheme of Pegg, Phillips and Barnett [Phys. Rev. Lett. 81, 1604 (1998)] -- the so-called quantum scissors device. The second method corresponds to the truncation scheme of Leo\\'nski and Tana\\'s [Phys. Rev. A 49, R20 (1994)]. We propose some new definitions of the states related to these truncation schemes and find their explicit forms in the Fock representation. We discuss finite-dimensional generalizations of coherent states, phase coherent states, displaced number states, Schr\\"odinger cats, and squeezed vacuum. We show some i...
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.)
Brothers, Harlan J.
2015-03-01
Benoit Mandelbrot always had a strong feeling that music could be viewed from a fractal perspective. However, without our eyes to guide us, how do we gain this perspective? Here we discuss precisely what it means to say that a piece of music is fractal.
Some New Exact Solutions to the Dispersive Long-Wave Equation in(2+1)-Dimensional Spaces
Institute of Scientific and Technical Information of China (English)
LI De-Sheng; ZHANG Hong-Qing
2003-01-01
In this paper, by using a further extended tanh method and symbolic computation system, some newsoliton-like and period form solutions of the dispersive long-wave equation in (2+1)-dimensional spaces are obtained.
Contour fractal analysis of grains
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.
The role of sleep in forming a memory representation of a two-dimensional space.
Coutanche, Marc N; Gianessi, Carol A; Chanales, Avi J H; Willison, Kate W; Thompson-Schill, Sharon L
2013-12-01
There is ample evidence from human and animal models that sleep contributes to the consolidation of newly learned information. The precise role of sleep for integrating information into interconnected memory representations is less well understood. Building on prior findings that following sleep (as compared to wakefulness) people are better able to draw inferences across learned associations in a simple hierarchy, we ask how sleep helps consolidate relationships in a more complex representational space. We taught 60 subjects spatial relationships between pairs of buildings, which (unknown to participants) formed a two-dimensional grid. Critically, participants were only taught a subset of the many possible spatial relations, which allowed them to potentially infer the remainder. After a 12 h period that either did or did not include a normal period of sleep, participants returned to the lab. We examined the quality of each participant's map of the two-dimensional space, and their knowledge of relative distances between buildings. After 12 h with sleep, subjects could more accurately map the full space than subjects who experienced only wakefulness. The incorporation of untaught, but inferable, associations was particularly improved. We further found that participants' distance judgment performance related to self-reported navigational style, but only after sleep. These findings demonstrate that consolidation over a night of sleep begins to integrate relations into an interconnected complex representation, in a way that supports spatial relational inference.
New Clustering Method in High-Dimensional Space Based on Hypergraph-Models
Institute of Scientific and Technical Information of China (English)
CHEN Jian-bin; WANG Shu-jing; SONG Han-tao
2006-01-01
To overcome the limitation of the traditional clustering algorithms which fail to produce meanirigful clusters in high-dimensional, sparseness and binary value data sets, a new method based on hypergraph model is proposed. The hypergraph model maps the relationship present in the original data in high dimensional space into a hypergraph. A hyperedge represents the similarity of attribute-value distribution between two points. A hypergraph partitioning algorithm is used to find a partitioning of the vertices such that the corresponding data items in each partition are highly related and the weight of the hyperedges cut by the partitioning is minimized. The quality of the clustering result can be evaluated by applying the intra-cluster singularity value.Analysis and experimental results have demonstrated that this approach is applicable and effective in wide ranging scheme.
Note: Silicon Carbide Telescope Dimensional Stability for Space-based Gravitational Wave Detectors
Sanjuah, J.; Korytov, D.; Mueller, G.; Spannagel, R.; Braxmaier, C.; Preston, A.; Livas, J.
2012-01-01
Space-based gravitational wave detectors are conceived to detect gravitational waves in the low frequency range by measuring the distance between proof masses in spacecraft separated by millions of kilometers. One of the key elements is the telescope which has to have a dimensional stability better than 1 pm Hz(exp -1/2) at 3 mHz. In addition, the telescope structure must be light, strong, and stiff. For this reason a potential telescope structure consisting of a silicon carbide quadpod has been designed, constructed, and tested. We present dimensional stability results meeting the requirements at room temperature. Results at -60 C are also shown although the requirements are not met due to temperature fluctuations in the setup.
Constructing Two-Dimensional Voronoi Diagrams via Divide-and-Conquer of Envelopes in Space
Setter, Ophir
2009-05-01
We present a general framework for computing two-dimensional Voronoi diagrams of different classes of sites under various distance functions. The framework is sufficiently general to support diagrams embedded on a family of two-dimensional parametric surfaces in $R^3$. The computation of the diagrams is carried out through the construction of envelopes of surfaces in 3-space provided by CGAL (the Computational Geometry Algorithm Library). The construction of the envelopes follows a divide-and-conquer approach. A straightforward application of the divide-and-conquer approach for computing Voronoi diagrams yields algorithms that are inefficient in the worst case. We prove that through randomization the expected running time becomes near-optimal in the worst case. We show how to employ our framework to realize various types of Voronoi diagrams with different properties by providing implementations for a vast collection of commonly used Voronoi diagrams. We also show how to apply the new framework and other exist...
Chudecki, Adam
2016-01-01
4-dimensional spaces equipped with 2-dimensional (complex holomorphic or real smooth) completely integrable distributions are considered. The integral manifolds of such distributions are totally null and totally geodesics 2-dimensional surfaces which are called the null strings. Properties of congruences (foliations) of such 2-surfaces are studied. Relation between properties of congruences of null strings and Petrov-Penrose type of SD Weyl spinor and algebraic types of traceless Ricci tensor is analyzed.
Vialidad, conectividad y fractales
Pineda Paz, Eduardo; Guerrero Torrenegra, Alejandro
2014-01-01
La morfología urbana es posible analizarla mediante ecuaciones no lineales que aparentemente reflejan el comportamiento del hombre. La teoría del caos, la incertidumbre y los fractales, aportan nuevas posibilidades al planificador urbano. El estudio es descriptivo y analítico, siguiendo pautas fenomenológicas, combinando teoría y práctica urbanística, con matemática sencilla. La parroquia Olegario Villalobos de Maracaibo es el caso de estudio. La investigación abordó la dimensión ...
Fractal apertures in waveguides, conducting screens and cavities analysis and design
Ghosh, Basudeb; Kartikeyan, M V
2014-01-01
This book deals with the design and analysis of fractal apertures in waveguides, conducting screens and cavities using numerical electromagnetics and field-solvers. The aim is to obtain design solutions with improved accuracy for a wide range of applications. To achieve this goal, a few diverse problems are considered. The book is organized with adequate space dedicated for the design and analysis of fractal apertures in waveguides, conducting screens, and cavities, microwave/millimeter wave applications followed by detailed case-study problems to infuse better insight and understanding of the subject. Finally, summaries and suggestions are given for future work. Fractal geometries were widely used in electromagnetics, specifically for antennas and frequency selective surfaces (FSS). The self-similarity of fractal geometry gives rise to a multiband response, whereas the space-filling nature of the fractal geometries makes it an efficient element in antenna and FSS unit cell miniaturization. Until now, no e...
Spatial Dynamics of Urban Growth Based on Entropy and Fractal Dimension
Chen, Yanguang
2016-01-01
The fractal dimension growth of urban form can be described with sigmoid functions such as logistic function due to squashing effect. The sigmoid curves of fractal dimension suggest a type of spatial replacement dynamics of urban evolution. How to understand the underlying rationale of the fractal dimension curves is a pending problem. This study is based on two previous findings. First, normalized fractal dimension proved to equal normalized spatial entropy; second, a sigmoid function proceeds from an urban-rural interaction model. Defining urban space-filling measurement by spatial entropy, and defining rural space-filling measurement by information gain, we can construct a new urban-rural interaction and coupling model. From this model, we can derive the logistic equation of fractal dimension growth strictly. This indicates that urban growth results from the unity of opposites between spatial entropy increase and information increase. In a city, an increase in spatial entropy is accompanied by a decrease i...
Navigation performance in virtual environments varies with fractal dimension of landscape.
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.
Study of desorption in a vapor dominated reservoir with fractal geometry
Energy Technology Data Exchange (ETDEWEB)
Tudor, Monica; Horne, Roland N.; Hewett, Thomas A.
1995-01-26
This paper is an attempt to model well decline in a vapor dominated reservoir with fractal geometry. The fractal network of fractures is treated as a continuum with characteristic anomalous diffusion of pressure. A numerical solver is used to obtain the solution of the partial differential equation including adsorption in the fractal storage space. The decline of the reservoir is found to obey the empirical hyperbolic type relation when adsorption is not present. Desorption does not change the signature of the flow rate decline but shifts it on the time/flow rate axis. Only three out of six model parameters can be estimated from field data, due to the linear correlation between parameters. An application to real well data from The Geysers field is presented together with the estimated reservoir, fractal space and adsorption parameters. Desorption dominated flow is still a questionable approximation for flow in fractal objects.
Directory of Open Access Journals (Sweden)
Mohammad Shahzad
2016-05-01
Full Text Available This study deals with the control of chaotic dynamics of tumor cells, healthy host cells, and effector immune cells in a chaotic Three Dimensional Cancer Model (TDCM by State Space Exact Linearization (SSEL technique based on Lie algebra. A non-linear feedback control law is designed which induces a coordinate transformation thereby changing the original chaotic TDCM system into a controlled one linear system. Numerical simulation has been carried using Mathematica that witness the robustness of the technique implemented on the chosen chaotic system.
Energy Technology Data Exchange (ETDEWEB)
Castellani, Marco; Giuli, Massimiliano, E-mail: massimiliano.giuli@univaq.it [University of L’Aquila, Department of Information Engineering, Computer Science and Mathematics (Italy)
2016-02-15
We study pseudomonotone and quasimonotone quasivariational inequalities in a finite dimensional space. In particular we focus our attention on the closedness of some solution maps associated to a parametric quasivariational inequality. From this study we derive two results on the existence of solutions of the quasivariational inequality. On the one hand, assuming the pseudomonotonicity of the operator, we get the nonemptiness of the set of the classical solutions. On the other hand, we show that the quasimonoticity of the operator implies the nonemptiness of the set of nonzero solutions. An application to traffic network is also considered.
Indian Academy of Sciences (India)
S Panda; B K Panda
2010-09-01
Chemical potential and internal energy of a noninteracting Fermi gas at low temperature are evaluated using the Sommerfeld method in the fractional-dimensional space. When temperature increases, the chemical potential decreases below the Fermi energy for any dimension equal to 2 and above due to the small entropy, while it increases above the Fermi energy for dimensions below 2 as a result of high entropy. The ranges of validity of the truncated series expansions of these quantities are extended from low to intermediate temperature regime as well as from high to relatively low density regime by using the Pad ́e approximant technique.
Canonical Formulation of A Bosonic Matter Field in 1+1 Dimensional Curved Space
Ghalati, R N; Sherry, T N
2006-01-01
We study a Bosonic scalar in 1+1 dimensional curved space that is coupled to a dynamical metric field. This metric, along with the affine connection, also appears in the Einstein-Hilbert action when written in first order form. After illustrating the Dirac constraint analysis in Yang-Mills theory, we apply this formulation to the Einstein-Hilbert action and the action of the Bosonic scalar field, first separately and then together. Only in the latter case does a dynamical degree of freedom emerge.
From High Dimensional Chaos to Stable Periodic Orbits: The Structure of Parameter Space
Energy Technology Data Exchange (ETDEWEB)
Barreto, E.; Hunt, B.R.; Grebogi, C.; Yorke, J.A. [University of Maryland, College Park, Maryland 20742 (United States)
1997-06-01
Regions in the parameter space of chaotic systems that correspond to stable behavior are often referred to as windows. In this Letter, we elucidate the occurrence of such regions in higher dimensional chaotic systems. We describe the fundamental structure of these windows, and also indicate under what circumstances one can expect to find them. These results are applicable to systems that exhibit several positive Lyapunov exponents, and are of importance to both the theoretical and the experimental understanding of dynamical systems. {copyright} {ital 1997} {ital The American Physical Society}
Dimensional regularization in position space and a Forest Formula for Epstein-Glaser renormalization
Dütsch, Michael; Fredenhagen, Klaus; Keller, Kai Johannes; Rejzner, Katarzyna
2014-12-01
We reformulate dimensional regularization as a regularization method in position space and show that it can be used to give a closed expression for the renormalized time-ordered products as solutions to the induction scheme of Epstein-Glaser. This closed expression, which we call the Epstein-Glaser Forest Formula, is analogous to Zimmermann's Forest Formula for BPH renormalization. For scalar fields, the resulting renormalization method is always applicable, we compute several examples. We also analyze the Hopf algebraic aspects of the combinatorics. Our starting point is the Main Theorem of Renormalization of Stora and Popineau and the arising renormalization group as originally defined by Stückelberg and Petermann.
String cosmological models in the Brans-Dicke theory for five-dimensional space-time
Institute of Scientific and Technical Information of China (English)
Koijam Manihar Singh; Kangujam Priyokumar Singh
2012-01-01
Five-dimensional space-time string cosmological models generated by a cloud of strings with particles attached to them are studied in the Brans-Dicke theory.We obtain two types of interesting models by taking up the cases of geometric strings (or Nambu strings) and p-strings (Takabayasi strings),and study their different physical and dynamical properties.The roles of the scalar field in getting different phases,such as the inflationary phase and the string-dominated phase,are discussed.An interesting feature obtained here is that in one of the models there is a "bounce" at a particular instant of its evolution.
Bhar, Piyali
2014-01-01
In this paper we are interested to search whether wormhole solutions exists in different dimensional noncommutative inspired spacetime.It is well known that noncommutativity of the space is an outcome of the string theory and it replaced the usual point like object by a smeared object.Here we have chosen Lorentzian distribution as the density function in the noncommutative inspired spacetime.We have observed that the wormhole solution exists only in four and five dimension,however higher than five dimension no wormhole exists.
Bifurcations of Fine 3-point-loop in Higher Dimensional Space
Institute of Scientific and Technical Information of China (English)
Yin Lai JIN; De Ming ZHU
2005-01-01
By using the linear independent fundamental solutions of the linear variational equation along the heteroclinic loop to establish a suitable local coordinate system in some small tubular neigha fine 3-point loop in higher dimensional space. Under some transversal conditions and the non-twisted condition, the existence, coexistence and incoexistence of 2-point-loop, 1-homoclinic orbit, simple 1-periodic orbit and 2-fold 1-periodic orbit, and the number of 1-periodic orbits are studied. Moreover,the bifurcation surfaces and existence regions are given. Lastly, the above bifurcation results are applied to a planar system and an inside stability criterion is obtained.
Energy Technology Data Exchange (ETDEWEB)
Keller, Kai Johannes
2010-04-15
The present work contains a consistent formulation of the methods of dimensional regularization (DimReg) and minimal subtraction (MS) in Minkowski position space. The methods are implemented into the framework of perturbative Algebraic Quantum Field Theory (pAQFT). The developed methods are used to solve the Epstein-Glaser recursion for the construction of time-ordered products in all orders of causal perturbation theory. A solution is given in terms of a forest formula in the sense of Zimmermann. A relation to the alternative approach to renormalization theory using Hopf algebras is established. (orig.)
Chaotic mixing and fractals in a geophysical jet current
Budyansky, M V; 10.1016/j.cnsns.2006.05.004
2012-01-01
We model Lagrangian lateral mixing and transport of passive scalars in meandering oceanic jet currents by two-dimensional advection equations with a kinematic stream function with a time-dependent amplitude of a meander imposed. The advection in such a model is known to be chaotic in a wide range of the meander's characteristics. We study chaotic transport in a stochastic layer and show that it is anomalous. The geometry of mixing is examined and shown to be fractal-like. The scattering characteristics (trapping time of advected particles and the number of their rotations around elliptical points) are found to have a hierarchical fractal structure as functions of initial particle's positions. A correspondence between the evolution of material lines in the flow and elements of the fractal is established.
Fractal Dimension of Certain Continuous Functions of Unbounded Variation
Liang, Y. S.; Su, W. Y.
Continuous functions on closed intervals are composed of bounded variation functions and unbounded variation functions. Fractal dimension of continuous functions with bounded variation must be one-dimensional (1D). While fractal dimension of continuous functions with unbounded variation may be 1 or not. Certain continuous functions of unbounded variation whose fractal dimensions are 1 have been mainly investigated in the paper. A continuous function on a closed interval with finite unbounded variation points has been proved to be 1D. Furthermore, we deal with continuous functions which have infinite unbounded variation points and part of them have been proved to be 1D. Certain examples of 1D continuous functions which have uncountable unbounded variation points have been given in the present paper.
Chaos, solitons and fractals in hidden symmetry models
Energy Technology Data Exchange (ETDEWEB)
Maccari, Attilio [Technical Institute ' G. Cardano' , Piazza della Resistenza 1, 00015 Monterotondo, Rome (Italy)] e-mail: solitone@yahoo.it
2006-01-01
A spontaneous symmetry breaking (or hidden symmetry) model is reduced to a system nonlinear evolution equations integrable via an appropriate change of variables, by means of the asymptotic perturbation (AP) method, based on spatio-temporal rescaling and Fourier expansion. It is demonstrated the existence of coherent solutions as well as chaotic and fractal patterns, due to the possibility of selecting appropriately some arbitrary functions. Dromion, lump, breather, instanton and ring soliton solutions are derived and the interaction between these coherent solutions are completely elastic, because they pass through each other and preserve their shapes and velocities, the only change being a phase shift. Finally, one can construct lower dimensional chaotic patterns such as chaotic-chaotic patterns, periodic-chaotic patterns, chaotic soliton and dromion patterns. In a similar way, fractal dromion and lump patterns as well as stochastic fractal excitations can appear in the solution.
Exploring High-Dimensional Data Space: Identifying Optimal Process Conditions in Photovoltaics
Energy Technology Data Exchange (ETDEWEB)
Suh, C.; Biagioni, D.; Glynn, S.; Scharf, J.; Contreras, M. A.; Noufi, R.; Jones, W. B.
2011-01-01
We demonstrate how advanced exploratory data analysis coupled to data-mining techniques can be used to scrutinize the high-dimensional data space of photovoltaics in the context of thin films of Al-doped ZnO (AZO), which are essential materials as a transparent conducting oxide (TCO) layer in CuIn{sub x}Ga{sub 1-x}Se{sub 2} (CIGS) solar cells. AZO data space, wherein each sample is synthesized from a different process history and assessed with various characterizations, is transformed, reorganized, and visualized in order to extract optimal process conditions. The data-analysis methods used include parallel coordinates, diffusion maps, and hierarchical agglomerative clustering algorithms combined with diffusion map embedding.
Energy Technology Data Exchange (ETDEWEB)
Suh, C.; Glynn, S.; Scharf, J.; Contreras, M. A.; Noufi, R.; Jones, W. B.; Biagioni, D.
2011-07-01
We demonstrate how advanced exploratory data analysis coupled to data-mining techniques can be used to scrutinize the high-dimensional data space of photovoltaics in the context of thin films of Al-doped ZnO (AZO), which are essential materials as a transparent conducting oxide (TCO) layer in CuInxGa1-xSe2 (CIGS) solar cells. AZO data space, wherein each sample is synthesized from a different process history and assessed with various characterizations, is transformed, reorganized, and visualized in order to extract optimal process conditions. The data-analysis methods used include parallel coordinates, diffusion maps, and hierarchical agglomerative clustering algorithms combined with diffusion map embedding.
Positioning with stationary emitters in a two-dimensional space-time
Coll, B; Morales, J A; Coll, Bartolom\\'{e}; Ferrando, Joan Josep; Morales, Juan Antonio
2006-01-01
The basic elements of the relativistic positioning systems in a two-dimensional space-time have been introduced in a previous work [Phys. Rev. D {\\bf 73}, 084017 (2006)] where geodesic positioning systems, constituted by two geodesic emitters, have been considered in a flat space-time. Here, we want to show in what precise senses positioning systems allow to make {\\em relativistic gravimetry}. For this purpose, we consider stationary positioning systems, constituted by two uniformly accelerated emitters separated by a constant distance, in two different situations: absence of gravitational field (Minkowski plane) and presence of a gravitational mass (Schwarzschild plane). The physical coordinate system constituted by the electromagnetic signals broadcasting the proper time of the emitters are the so called {\\em emission coordinates}, and we show that, in such emission coordinates, the trajectories of the emitters in both situations, absence and presence of a gravitational field, are identical. The interesting...
Patricio, Pedro; Duarte, Jorge; Januario, Cristina
2015-01-01
We investigate the rheology of a fractal network, in the framework of the linear theory of viscoelasticity. We identify each segment of the network with a simple Kelvin-Voigt element, with a well defined equilibrium length. The final structure retains the elastic characteristics of a solid or a gel. By considering a very simple regular self-similar structure of segments in series and in parallel, in 1, 2 or 3 dimensions, we are able to express the viscoelasticity of the network as an effective generalised Kelvin-Voigt model with a power law spectrum of retardation times, $\\phi\\sim\\tau^{\\alpha-1}$. We relate the parameter $\\alpha$ with the fractal dimension of the gel. In some regimes ($0<\\alpha<1$), we recover the weak power law behaviours of the elastic and viscous moduli with the angular frequencies, $G'\\sim G''\\sim w^\\alpha$, that occur in a variety of soft materials, including living cells. In other regimes, we find different and interesting power laws for $G'$ and $G''$.
Directory of Open Access Journals (Sweden)
Ai-Min Yang
2014-03-01
Full Text Available The fractal heat flow within local fractional derivative is investigated. The nonhomogeneous heat equations arising in fractal heat flow are discussed. The local fractional Fourier series solutions for one-dimensional nonhomogeneous heat equations are obtained. The nondifferentiable series solutions are given to show the efficiency and implementation of the present method.
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WANG Ling
2007-08-01
Full Text Available The solidification microstructure and fractal characteristics of the solid-liquid interfaces of Inconel 718, under different cooling rates during directional solidification, were investigated by using SEM. Results showed that 5 μm/s was the cellular-dendrite transient rate. The prime dendrite arm spacing (PDAS was measured by Image Tool and it decreased with the cooling rate increased. The fractal dimension of the interfaces was calculated and it changes from 1.204310 to 1.517265 with the withdrawal rate ranging from 10 to 100 μm/s. The physical significance of the fractal dimension was analyzed by using fractal theory. It was found that the fractal dimension of the dendrites can be used to describe the solidification microstructure and parameters at low cooling rate, but both the fractal dimension and the dendrite arm spacing are needed in order to integrally describe the evaluation of the solidification microstructure completely.
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
The solidification microstructure and fractal characteristics of the solid-liquid interfaces of Inconel718, under different cooling rates during directional solidification, ware investigated by using SEM. Results showed that 5 μm/s was the cellular-dendrite transient rate. The prime dendrite arm spacing (PDAS) was measured by Image Tool and it decreased with the cooling rate increased. The fractal dimension of the interfaces was calculated and it changes from 1.204310 to 1.517265 with the withdrawal rate ranging from 10 to 100 μm/s. The physical significance of the fractal dimension was analyzed by using fractal theory. It was found that the fractal dimension of the dendrites can be used to describe the solidification microstructure and parameters at low cooling rate, but both the fractal dimension and the dendrite arm spacing are needed in order to integrally describe the evaluation of the solidification microstructure completely.
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Javier Rodríguez
2005-09-01
Full Text Available La geometría fractal caracteriza de manera adecuada las formas irregulares de la naturaleza; por ello, para obtener una descripción matemática objetiva de la irregularidad de la cavidad torácica, es necesario hacer uso de ésta. Mediante el método de box-counting se midieron las dimensiones fractales de los objetos definidos de 20 radiografías de tórax de individuos sanos, 10 de mujeres y 10 de hombres entre 18 y 32 años de edad, en la cuales se comparó la primera y segunda cifra significativa entre las dimensiones fractales de los tres objetos definidos. Las dimensiones fractales de las placas evaluadas con la primera cifra significativa, resultaron diferentes o iguales en los tres objetos evaluados, o bien iguales para dos de los tres objetos fractales, mientras que en la segunda cifra significativa fueron diferentes en los tres objetos fractales al mismo tiempo, además la mayor diferencia máxima entre los valores de las islas es de tres décimas. Estudios posteriores con radiografías de pacientes con diferentes patologías permitirán realizar una generalización de la metodología desarrollada para su aplicación clínica.Fractal geometry measures in a proper way the irregular forms of nature. For this reason, we must use it in order to obtain an objective mathematical description of the thoracic cavity’s irregularity. By means of the box counting method, fractal dimensions of defined objects of 20 thoracic X-rays from healthy individuals were measured. 10 were men and 10 were women, between 18 and 32 years. A comparison was made between the first and the second significant cipher among the fractal dimensions of the three defined objects. The fractal dimensions of the radiographies evaluated with the first significant cipher turned out to be different or equal in the three evaluated objects or equal for two of the three fractal objects, while in the second significant cipher these were different in the three fractal objects at
Fractal analysis of sulphidic mineral
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Miklúová Viera
2002-03-01
Full Text Available In this paper, the application of fractal theory in the characterization of fragmented surfaces, as well as the mass-size distributions are discussed. The investigated mineral-chalcopyrite of Slovak provenience is characterised after particle size reduction processes-crushing and grinding. The problem how the different size reduction methods influence the surface irregularities of obtained particles is solved. Mandelbrot (1983, introducing the fractal geometry, offered a new way of characterization of surface irregularities by the fractal dimension. The determination of the surface fractal dimension DS consists in measuring the specific surface by the BET method in several fractions into which the comminuted chalcopyrite is sieved. This investigation shows that the specific surface of individual fractions were higher for the crushed sample than for the short-term (3 min ground sample. The surface fractal dimension can give an information about the adsorption sites accessible to molecules of nitrogen and according to this, the value of the fractal dimension is higher for crushed sample.The effect of comminution processes on the mass distribution of particles crushed and ground in air as well as in polar liquids is also discussed. The estimation of fractal dimensions of particles mass distribution is done on the assumption that the particle size distribution is described by the power-law (1. The value of fractal dimension for the mass distribution in the crushed sample is lower than in the sample ground in air, because it is influenced by the energy required for comminution.The sample of chalcopyrite was ground (10min in ethanol and i-butanol [which according to Ikazaki (1991] are characterized by the parameter µ /V, where µ is its dipole moment and V is the molecular volume. The values of µ /V for the used polar liquids are of the same order. That is why the expressive differences in particle size distributions as well as in the values of