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Genzor J, Gendiar A, Nishino T. Local and global magnetization on the Sierpiński carpet. Phys Rev E 2023; 107:044108. [PMID: 37198768 DOI: 10.1103/physreve.107.044108] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/01/2023] [Accepted: 03/21/2023] [Indexed: 05/19/2023]
Abstract
The phase transition of the classical Ising model on the Sierpiński carpet, which has the fractal dimension log_{3}^{}8≈1.8927, is studied by an adapted variant of the higher-order tensor renormalization group method. The second-order phase transition is observed at the critical temperature T_{c}^{}≈1.478. Position dependence of local functions is studied through impurity tensors inserted at different locations on the fractal lattice. The critical exponent β associated with the local magnetization varies by two orders of magnitude, depending on lattice locations, whereas T_{c}^{} is not affected. Furthermore, we employ automatic differentiation to accurately and efficiently compute the average spontaneous magnetization per site as a first derivative of free energy with respect to the external field, yielding the global critical exponent of β≈0.135.
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Affiliation(s)
- Jozef Genzor
- Department of Physics, Graduate School of Science, Kobe University, Kobe 657-8501, Japan
- Physics Division, National Center for Theoretical Sciences, Taipei 10617, Taiwan
- Physics Division, National Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan
| | - Andrej Gendiar
- Institute of Physics, Slovak Academy of Sciences, Dúbravská Cesta 9, SK-845 11 Bratislava, Slovakia
| | - Tomotoshi Nishino
- Department of Physics, Graduate School of Science, Kobe University, Kobe 657-8501, Japan
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2
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Abstract
It is a central prediction of renormalization group theory that the critical behaviours of many statistical mechanics models on Euclidean lattices depend only on the dimension and not on the specific choice of lattice. We investigate the extent to which this universality continues to hold beyond the Euclidean setting, taking as case studies Bernoulli bond percolation and lattice trees. We present strong numerical evidence that the critical exponents governing these models on transitive graphs of polynomial volume growth depend only on the volume-growth dimension of the graph and not on any other large-scale features of the geometry. For example, our results strongly suggest that percolation, which has upper-critical dimension 6, has the same critical exponents on
Z
4
and the Heisenberg group despite the distinct large-scale geometries of these two lattices preventing the relevant percolation models from sharing a common scaling limit. On the other hand, we also show that no such universality should be expected to hold on fractals, even if one allows the exponents to depend on a large number of standard fractal dimensions. Indeed, we give natural examples of two fractals which share Hausdorff, spectral, topological and topological Hausdorff dimensions but exhibit distinct numerical values of the percolation Fisher exponent
τ
. This gives strong evidence against a conjecture of Balankin
et al.
(2018
Phys. Lett. A
382
, 12–19 (
doi:10.1016/j.physleta.2017.10.035
)).
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Affiliation(s)
- Noah Halberstam
- Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK
| | - Tom Hutchcroft
- Division of Physics, Mathematics and Astronomy, California Institute of Technology, Pasadena, CA, USA
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Genzor J, Gendiar A, Kao YJ. J_{1}-J_{2} fractal studied by multirecursion tensor-network method. Phys Rev E 2022; 105:024124. [PMID: 35291084 DOI: 10.1103/physreve.105.024124] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/23/2021] [Accepted: 01/28/2022] [Indexed: 06/14/2023]
Abstract
We generalize a tensor-network algorithm to study the thermodynamic properties of self-similar spin lattices constructed on a square-lattice frame with two types of couplings, J_{1}^{} and J_{2}^{}, chosen to transform a regular square lattice (J_{1}^{}=J_{2}^{}) onto a fractal lattice if decreasing J_{2}^{} to zero (the fractal fully reconstructs when J_{2}^{}=0). We modified the higher-order tensor renormalization group (HOTRG) algorithm for this purpose. Single-site measurements are performed by means of so-called impurity tensors. So far, only a single local tensor and uniform extension-contraction relations have been considered in HOTRG. We introduce 10 independent local tensors, each being extended and contracted by 15 different recursion relations. We applied the Ising model to the J_{1}^{}-J_{2}^{} planar fractal whose Hausdorff dimension at J_{2}^{}=0 is d^{(H)}=ln12/ln4≈1.792. The generalized tensor-network algorithm is applicable to a wide range of fractal patterns and is suitable for models without translational invariance.
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Affiliation(s)
- Jozef Genzor
- Department of Physics, National Taiwan University, Taipei 10607, Taiwan
| | - Andrej Gendiar
- Institute of Physics, Slovak Academy of Sciences, SK-845 11 Bratislava, Slovakia
| | - Ying-Jer Kao
- Department of Physics, National Taiwan University, Taipei 10607, Taiwan
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4
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de Bruijn R, van der Schoot P. Connectedness percolation of fractal liquids. Phys Rev E 2021; 104:054605. [PMID: 34942762 DOI: 10.1103/physreve.104.054605] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/04/2021] [Accepted: 10/15/2021] [Indexed: 11/07/2022]
Abstract
We apply connectedness percolation theory to fractal liquids of hard particles, and make use of a Percus-Yevick liquid state theory combined with a geometric connectivity criterion. We find that in fractal dimensions the percolation threshold interpolates continuously between integer-dimensional values, and that it decreases monotonically with increasing (fractal) dimension. The influence of hard-core interactions is significant only for dimensions below three. Finally, our theory incorrectly suggests that a percolation threshold is absent below about two dimensions, which we attribute to the breakdown of the connectedness Percus-Yevick closure.
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Affiliation(s)
- René de Bruijn
- Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.,Institute for Complex Molecular Systems, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
| | - Paul van der Schoot
- Department of Applied Physics, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
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Duran-Nebreda S, Johnston IG, Bassel GW. Efficient vasculature investment in tissues can be determined without global information. J R Soc Interface 2020; 17:20200137. [PMID: 32316879 PMCID: PMC7211487 DOI: 10.1098/rsif.2020.0137] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/26/2020] [Accepted: 03/24/2020] [Indexed: 12/28/2022] Open
Abstract
Cells are the fundamental building blocks of organs and tissues. Information and mass flow through cellular contacts in these structures is vital for the orchestration of organ function. Constraints imposed by packing and cell immobility limit intercellular communication, particularly as organs and organisms scale up to greater sizes. In order to transcend transport limitations, delivery systems including vascular and respiratory systems evolved to facilitate the movement of matter and information. The construction of these delivery systems has an associated cost, as vascular elements do not perform the metabolic functions of the organs they are part of. This study investigates a fundamental trade-off in vascularization in multicellular tissues: the reduction of path lengths for communication versus the cost associated with producing vasculature. Biologically realistic generative models, using multicellular templates of different dimensionalities, revealed a limited advantage to the vascularization of two-dimensional tissues. Strikingly, scale-free improvements in transport efficiency can be achieved even in the absence of global knowledge of tissue organization. A point of diminishing returns in the investment of additional vascular tissue to the increased reduction of path length in 2.5- and three-dimensional tissues was identified. Applying this theory to experimentally determined biological tissue structures, we show the possibility of a co-dependency between the method used to limit path length and the organization of cells it acts upon. These results provide insight as to why tissues are or are not vascularized in nature, the robustness of developmental generative mechanisms and the extent to which vasculature is advantageous in the support of organ function.
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Affiliation(s)
| | - Iain G Johnston
- Faculty of Mathematics and Natural Sciences, University of Bergen, Bergen, Norway
| | - George W Bassel
- School of Life Sciences, University of Warwick, Coventry CV4 7AL, UK
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6
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Perreau M. Critical temperatures of the Ising model on Sierpiñski fractal lattices. EPJ WEB OF CONFERENCES 2020. [DOI: 10.1051/epjconf/202024401013] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
We report our latest results of the spectra and critical temperatures of the partition function of the Ising model on deterministic Sierpiñski carpets in a wide range of fractal dimensions. Several examples of spectra are given. When the fractal dimension increases (and correlatively the lacunarity decreases), the spectra aggregates more and more tightly along the spectrum of the regular square lattice. The single real rootvc, comprised between 0 and 1, of the partition function, which corresponds to the critical temperatureTcthrough the formulavc= tanh(1/Tc), reliably fits a power law of exponentkwherekis the segmentation step of the fractal structure. This simple expression allows to extrapolate the critical temperature fork→ ∞. The plot of the logarithm of this extrapolated critical temperature versus the fractal dimension appears to be reliably linear in a wide range of fractal dimensions, except for highly lacunary structures of fractal dimensions close from 1 (the dimension of a quasilinear lattice) where the critical temperature goes to 0 and its logarithm to −∞.
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7
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Ay B, Mendes VC, Zhang L, Davies JE. A "best fit" approach for synergistic surface parameters to guide the design of candidate implant surfaces. J Biomed Mater Res B Appl Biomater 2019; 107:2165-2177. [PMID: 30677220 DOI: 10.1002/jbm.b.34312] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/23/2018] [Revised: 11/23/2018] [Accepted: 12/19/2018] [Indexed: 11/07/2022]
Abstract
Human bone resorption surfaces can provide a template for endosseous implant surface design. We characterized the topography of such sites using four synergistic parameters (fractal dimension, lacunarity, porosity, and surface roughness) and compared the generated values with those obtained from two groups of candidate titanium implant surfaces. For the first group (n = 5/group): grit-blasted acid etched (BAE), BAE with either discrete calcium phosphate crystal deposition or nanotube formation, machined titanium with nanotubes, or a nanofiber surface; each measured synergistic parameter was statistically compared with that of the resorbed bone surface and scored for inclusion in a "best fit" analysis. The analysis informed changes that could be made to a candidate implant surface to render it a closer "best fit" to that of the resorbed bone surface. In a second group of either titanium or titanium alloy implants their micro-topography, created by dual acid etching, was the same for each material substrate; but their nanotopographic complexity was changed by varying the degree of calcium phosphate crystalline deposits. These implants were also used in vivo where bone anchorage was tested using a tensile disruption test; and the "best fit" of synergistic parameters coincided with the best biological outcome for both titanium and titanium alloy implants. In conclusion, the four chosen synergistic parameters can be used to guide the sub-micron surface design of candidate implants, and our "best fit" approach is capable of identifying the surfaces with the best biological outcomes. © 2019 Wiley Periodicals, Inc. J Biomed Mater Res Part B: Appl Biomater 107B: 2165-2177, 2019.
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Affiliation(s)
- Birol Ay
- Institute of Biomaterials and Biomedical Engineering, University of Toronto, Toronto, Ontario, Canada
| | - Vanessa C Mendes
- Faculty of Dentistry, University of Toronto, Toronto, Ontario, Canada
| | - Li Zhang
- Analytical & Testing Center, Sichuan University, Chengdu, China
| | - John E Davies
- Institute of Biomaterials and Biomedical Engineering, University of Toronto, Toronto, Ontario, Canada.,Faculty of Dentistry, University of Toronto, Toronto, Ontario, Canada
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8
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Genzor J, Gendiar A, Nishino T. Phase transition of the Ising model on a fractal lattice. Phys Rev E 2016; 93:012141. [PMID: 26871057 DOI: 10.1103/physreve.93.012141] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/18/2015] [Indexed: 06/05/2023]
Abstract
The phase transition of the Ising model is investigated on a planar lattice that has a fractal structure. On the lattice, the number of bonds that cross the border of a finite area is doubled when the linear size of the area is extended by a factor of 4. The free energy and the spontaneous magnetization of the system are obtained by means of the higher-order tensor renormalization group method. The system exhibits the order-disorder phase transition, where the critical indices are different from those of the square-lattice Ising model. An exponential decay is observed in the density-matrix spectrum even at the critical point. It is possible to interpret that the system is less entangled because of the fractal geometry.
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Affiliation(s)
- Jozef Genzor
- Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, SK-845 11 Bratislava, Slovakia
| | - Andrej Gendiar
- Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, SK-845 11 Bratislava, Slovakia
| | - Tomotoshi Nishino
- Department of Physics, Graduate School of Science, Kobe University, Kobe 657-8501, Japan
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9
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Corberi F, Zannetti M, Lippiello E, Burioni R, Vezzani A. Phase ordering in disordered and inhomogeneous systems. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:062122. [PMID: 26172676 DOI: 10.1103/physreve.91.062122] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/26/2015] [Indexed: 06/04/2023]
Abstract
We study numerically the coarsening dynamics of the Ising model on a regular lattice with random bonds and on deterministic fractal substrates. We propose a unifying interpretation of the phase-ordering processes based on two classes of dynamical behaviors characterized by different growth laws of the ordered domain size, namely logarithmic or power law, respectively. It is conjectured that the interplay between these dynamical classes is regulated by the same topological feature that governs the presence or the absence of a finite-temperature phase transition.
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Affiliation(s)
- Federico Corberi
- Dipartimento di Fisica "E. R. Caianiello," and INFN, Gruppo Collegato di Salerno, and CNISM, Unità di Salerno, Università di Salerno, via Giovanni Paolo II 132, 84084 Fisciano (SA), Italy
| | - Marco Zannetti
- Dipartimento di Fisica "E. R. Caianiello," and INFN, Gruppo Collegato di Salerno, and CNISM, Unità di Salerno, Università di Salerno, via Giovanni Paolo II 132, 84084 Fisciano (SA), Italy
| | - Eugenio Lippiello
- Department of Mathematics and Physics, Second University of Naples, Viale Lincoln 5, 81100 Caserta, Italy
| | - Raffaella Burioni
- Dipartimento di Fisica e Scienza della Terra, and INFN, Gruppo Collegato di Parma, Università di Parma, Parco Area delle Scienze 7/A, I-423100 Parma, Italy
| | - Alessandro Vezzani
- Centro S3, CNR-Istituto di Nanoscienze, Via Campi 213A, 41125 Modena, Italy and Dipartimento di Fisica Scienza della Terra, Università di Parma, Parco Area delle Scienze 7/A, I-43100 Parma, Italy
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10
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Frisch U, Pomyalov A, Procaccia I, Ray SS. Turbulence in noninteger dimensions by fractal Fourier decimation. PHYSICAL REVIEW LETTERS 2012; 108:074501. [PMID: 22401207 DOI: 10.1103/physrevlett.108.074501] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/05/2011] [Indexed: 05/20/2023]
Abstract
Fractal decimation reduces the effective dimensionality D of a flow by keeping only a (randomly chosen) set of Fourier modes whose number in a ball of radius k is proportional to k(D) for large k. At the critical dimension D(c)=4/3 there is an equilibrium Gibbs state with a k(-5/3) spectrum, as in V. L'vov et al., Phys. Rev. Lett. 89, 064501 (2002). Spectral simulations of fractally decimated two-dimensional turbulence show that the inverse cascade persists below D=2 with a rapidly rising Kolmogorov constant, likely to diverge as (D-4/3)(-2/3).
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Affiliation(s)
- Uriel Frisch
- UNS, CNRS, OCA, Laboratoire Lagrange, Nice, France
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11
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Bab MA, Fabricius G, Albano EV. Revisiting random walks in fractal media: on the occurrence of time discrete scale invariance. J Chem Phys 2008; 128:044911. [PMID: 18248004 DOI: 10.1063/1.2823732] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
This paper addresses the kinetic behavior of random walks in fractal media. We perform extensive numerical simulations of both single and annihilating random walkers on several Sierpinski carpets, in order to study the time behavior of three observables: the average number of distinct sites visited by a single walker, the mean-square displacement from the origin, and the density of annihilating random walkers. We found that the time behavior of those observables is given by a power law modulated by soft logarithmic-periodic oscillations. We conjecture that logarithmic-periodic oscillations are a manifestation of a time domain discrete scale iNvariance (DSI) that occurs as a consequence of the spatial DSI of the substrate. Our conjecture implies that the logarithmic periods of oscillations in space and time domains are linked by a dynamic exponent z, through z=log(tau)/log(b(1)), where tau and b(1) are the fundamental scaling ratios of the DSI symmetry in the time and space domains, respectively. We use this relationship in order to compute z for different observables and fractals. Furthermore, we check the values obtained with independent measurements provided by the power-law behavior of the mean-square displacement with time [R(2)(t) proportional variant t(2/z)]. The very good agreement obtained between both computations of the z exponent gives strong support to the idea of an intimate interplay between spatial and time symmetry properties that we expect will have a quite general scope. We expect that the application of the outlined concepts in the field of dynamic processes in fractal media will stimulate further research.
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Affiliation(s)
- M A Bab
- Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas (INIFTA), UNLP, CONICET, Casilla de Correo 16, Sucursal 4, (1900) La Plata, Argentina
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12
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Fritsche M, Roman HE, Porto M. Temperature-dependent structural behavior of self-avoiding walks on Sierpinski carpets. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:061101. [PMID: 18233808 DOI: 10.1103/physreve.76.061101] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/06/2007] [Indexed: 05/25/2023]
Abstract
We study the temperature-dependent structural behavior of self-avoiding walks (SAWs) on two-dimensional Sierpinski carpets as a simple model of polymers adsorbed on a disordered surface. Thereby, the Sierpinski carpet defines two types of sites with energy 0 and >0 , respectively, yielding a deterministic fractal energy landscape. In the limiting cases of temperature T-->0 and T-->infinity , the known behaviors of SAWs on Sierpinski carpets and on regular square lattices, respectively, are recovered. For finite temperatures, the structural behavior is found to be intermediate between the two limiting cases; the characteristic exponents, however, display a nontrivial dependence on temperature.
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Affiliation(s)
- Miriam Fritsche
- Institut für Festkörperphysik, Technische Universität Darmstadt, Hochschulstrasse 8, 64289 Darmstadt, Germany
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13
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Lee SB, Kim YN. Absorbing phase transition in conserved lattice gas model on fractal lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:031137. [PMID: 17930229 DOI: 10.1103/physreve.76.031137] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/31/2007] [Indexed: 05/25/2023]
Abstract
We study the continuous phase transition of the conserved lattice gas (CLG) model from an active phase into an absorbing phase on two fractal lattices, i.e., on a checkerboard fractal and on a Sierpinski gasket. In the CLG model, a particle is assumed to be active if any of the neighboring sites are occupied by a particle and inactive if all neighboring sites are empty. We estimate critical exponents theta, beta, nu||, and nu perpendicular, characterizing, respectively, the density of active particles in time, the order parameter, the temporal and spatial correlation lengths near the critical point, and the results are confirmed by off-critical scaling and finite size scaling. The order parameter exponent beta on a checkerboard fractal appears to lie between the one-dimensional (1D) value and two-dimensional (2D) value of the CLG model, while that on a Sierpinski gasket lies between the 1D and 2D values of the conserved threshold transfer process. Such a difference is manifested based on the intrinsic properties of the underlying fractal lattices.
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Affiliation(s)
- Sang B Lee
- Department of Physics, Kyungpook National University, Taegu 702-701 Korea.
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14
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Marini F, Ordemann A, Porto M, Roman HE. Violation of the des Cloizeaux relation for self-avoiding walks on Sierpinski square lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:051102. [PMID: 17279872 DOI: 10.1103/physreve.74.051102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/2006] [Revised: 08/29/2006] [Indexed: 05/13/2023]
Abstract
The statistics of self-avoiding walks (SAWs) on deterministic fractal structures with infinite ramification, modeled by Sierpinski square lattices, is revisited in two and three dimensions using the reptation algorithm. The probability distribution function of the end-to-end distance of SAWs, consisting of up to 400 steps, is obtained and its scaling behavior at small distances is studied. The resulting scaling exponents are confronted with previous calculations for much shorter linear chains (20 to 30 steps) based on the exact enumeration (EE) technique. The present results coincide with the EE values in two dimensions, but differ slightly in three dimensions. A possible explanation for this discrepancy is discussed. Despite this, the violation of the so-called des Cloizeaux relation, a renormalization result that holds on regular lattices and on deterministic fractal structures with finite ramification, is confirmed numerically.
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Affiliation(s)
- Francesco Marini
- Dipartimento di Fisica, Università di Milano, Via Celoria 16, 20133 Milano, Italy
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15
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Bab MA, Fabricius G, Albano EV. Discrete scale invariance effects in the nonequilibrium critical behavior of the Ising magnet on a fractal substrate. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 74:041123. [PMID: 17155038 DOI: 10.1103/physreve.74.041123] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/09/2006] [Indexed: 05/12/2023]
Abstract
The nonequilibrium critical dynamics of the Ising magnet on a fractal substrate, namely the Sierpinski carpet with Hausdorff dimension d(H)=1.7925, has been studied within the short-time regime by means of Monte Carlo simulations. The evolution of the physical observables was followed at criticality, after both annealing ordered spin configurations (ground state) and quenching disordered initial configurations (high temperature state), for three segmentation steps of the fractal. We have obtained evidence showing that during these relaxation processes both the growth and the fragmentation of magnetic domains become influenced by the hierarchical structure of the substrate. In fact, the interplay between the dynamic behavior of the magnet and the underlying fractal leads to the emergence of a logarithmic-periodic oscillation, superimposed to a power law, which has been observed in the time dependence of both the decay of the magnetization and its logarithmic derivative. These oscillations have been carefully characterized in order to determine the critical temperature of the second-order phase transition and the critical exponents corresponding to the short-time regime. The effects of the substrate can also be observed from the dependence of the effective critical exponents on the segmentation step. The exponent theta of the initial increase of the magnetization has also been obtained and the results suggest that it would be almost independent of the fractal dimension of the substrate, provided that d(H) is close enough to d=2. The oscillations have been discussed within the framework of the discrete scale invariance of the substrate.
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Affiliation(s)
- M A Bab
- Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Sucursal 4, CC 16 (1900) La Plata, Argentina.
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16
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Yasar F, Akgünlü F. Fractal dimension and lacunarity analysis of dental radiographs. Dentomaxillofac Radiol 2005; 34:261-7. [PMID: 16120874 DOI: 10.1259/dmfr/85149245] [Citation(s) in RCA: 51] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/16/2023] Open
Abstract
OBJECTIVE As the occlusal forces transmitted to the jaw bones during mastication might be different in dentate and edentulous regions, there might be different radiographical trabecular bone texture in these regions. Image analysis procedures are promising techniques which are used to detect structural changes of bone texture on radiographs. In this study, the differences of fractal dimension (FD) and lacunarity measurements of radiographical trabecular bone between dentate and edentulous regions were investigated. METHODS Direct digital radiographs of premolar-molar region were taken from 51 patients who were included in our study. Two rectangular regions of interest (ROIs) with the same dimensions (37x119 pixels) were created on these radiographs; one in the edentulous region and the other one in the dentate region. The ROIs were segmented as black and white areas. Box-counting fractal dimension and lacunarity of these regions were calculated. RESULTS Paired samples t-test and Pearson correlation coefficients were calculated. It was found that there were differences between dentate and edentulous regions for FD and lacunarity (P=0.000). There is a negative correlation between FD and lacunarity (-0.643, P<0.01), positive correlation between dentate and edentulous regions and FD (0.819, P<0.01), and a negative correlation between lacunarity and dentate and edentulous regions (-0.541, P<0.01). CONCLUSIONS The differences of occlusal forces generated in dentate and edentulous regions during mastication cause some alterations in trabecular bone structure, and fractal dimension and lacunarity can reveal these alterations quantitatively.
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Affiliation(s)
- F Yasar
- Selcuk Universitesi Dis Hekimligi Fakultesi, Oral Diagnoz ve Radyoloji Anabilim Dali, 42075 Konya, Turkey.
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17
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Bab MA, Fabricius G, Albano EV. Critical behavior of an Ising system on the Sierpinski carpet: a short-time dynamics study. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 71:036139. [PMID: 15903525 DOI: 10.1103/physreve.71.036139] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/06/2004] [Revised: 10/29/2004] [Indexed: 05/02/2023]
Abstract
The short-time dynamic evolution of an Ising model embedded in an infinitely ramified fractal structure with noninteger Hausdorff dimension was studied using Monte Carlo simulations. Completely ordered and disordered spin configurations were used as initial states for the dynamic simulations. In both cases, the evolution of the physical observables follows a power-law behavior. Based on this fact, the complete set of critical exponents characteristic of a second-order phase transition was evaluated. Also, the dynamic exponent theta of the critical initial increase in magnetization, as well as the critical temperature, were computed. The exponent theta exhibits a weak dependence on the initial (small) magnetization. On the other hand, the dynamic exponent z shows a systematic decrease when the segmentation step is increased, i.e., when the system size becomes larger. Our results suggest that the effective noninteger dimension for the second-order phase transition is noticeably smaller than the Hausdorff dimension. Even when the behavior of the magnetization (in the case of the ordered initial state) and the autocorrelation (in the case of the disordered initial state) with time are very well fitted by power laws, the precision of our simulations allows us to detect the presence of a soft oscillation of the same type in both magnitudes that we attribute to the topological details of the generating cell at any scale.
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Affiliation(s)
- M A Bab
- Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas (INIFTA), Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Sucursal 4, Casilla de Correo 16, (1900) La Plata, Argentina
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18
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Mazumder S, Sen D, Patra AK, Khadilkar SA, Cursetji RM, Loidl R, Baron M, Rauch H. Dynamical scaling of the structure factor of some non-Euclidean systems. PHYSICAL REVIEW LETTERS 2004; 93:255704. [PMID: 15697913 DOI: 10.1103/physrevlett.93.255704] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/27/2004] [Indexed: 05/24/2023]
Abstract
Predictions of nonlinear theories on dynamics of new phase formation have been examined for the hydration of calcium silicates with light water and heavy water. In the case of hydration with light water, reasonable agreement has been observed with dynamical scaling hypothesis with a new measure of the characteristic length. The characteristic length does not follow a power law relation with time. Hydrating mass is found to be mass fractal throughout hydration, with mass fractal dimension increasing with time. But, in the case of hydration with heavy water, no agreement has been observed with the scaling hypothesis. Hydrating mass undergoes transition from mass fractal to surface fractal and finally again to mass fractal. The qualitative features of the kinetics of hydration, as measured in small-angle scattering experiments, are strikingly different for hydration with light water and heavy water.
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Affiliation(s)
- S Mazumder
- Solid State Physics Division, Bhabha Atomic Research Centre, Mumbai 400 085, India.
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19
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Li J, Nekka F. The Hausdorff measure functions: A new way to characterize fractal sets. Pattern Recognit Lett 2003. [DOI: 10.1016/s0167-8655(03)00115-6] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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20
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Vezzani A. Spontaneous magnetization of the Ising model on the Sierpinski carpet fractal, a rigorous result. ACTA ACUST UNITED AC 2003. [DOI: 10.1088/0305-4470/36/6/305] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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21
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Ordemann A, Porto M, Roman HE. Self-avoiding walks on self-similar structures: finite versus infinite ramification. ACTA ACUST UNITED AC 2002. [DOI: 10.1088/0305-4470/35/38/306] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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22
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Liaw TM, Huang MC, Chou YL, Lin SC. Evolution and structure formation of the distribution of partition function zeros: triangular type Ising lattices with cell decoration. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:066124. [PMID: 12188800 DOI: 10.1103/physreve.65.066124] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/06/2002] [Indexed: 05/23/2023]
Abstract
The distribution of partition function zeros of the two-dimensional Ising model in the complex temperature plane is studied within the context of triangular decorated lattices and their triangle-star transformations. Exact recursion relations for the zeros are deduced for the description of the evolution of the distribution of the zeros subject to the change of decoration level. In the limit of infinite decoration level, the decorated lattices essentially possess the Sierpiński gasket or its triangle-star transformation as the inherent structure. The positions of the zeros for the infinite decorated lattices are shown to coincide with the ones for the Sierpiński gasket or its triangle-star transformation, and the distributions of zeros all appear to be a union of infinite scattered points and a Jordan curve, which is the limit of the scattered points.
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Affiliation(s)
- Tsong-Ming Liaw
- Department of Physics, Chung-Yuan Christian University, Chungli 320, Taiwan
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23
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Ball RC, Caldarelli G, Flammini A. Angular structure of lacunarity, and the renormalization group. PHYSICAL REVIEW LETTERS 2000; 85:5134-5137. [PMID: 11102204 DOI: 10.1103/physrevlett.85.5134] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/24/2000] [Indexed: 05/23/2023]
Abstract
We formulate the angular structure of lacunarity in fractals, in terms of a symmetry reduction of the three point correlation function. This provides a rich probe of universality, and first measurements yield new evidence in support of the equivalence between self-avoiding walks (SAW's) and percolation perimeters in two dimensions. We argue that the lacunarity reveals much of the renormalization group in real space. This is supported by exact calculations for random walks and measured data for percolation clusters and SAW's. Relationships follow between exponents governing inward and outward propagating perturbations, and we also find a very general test for the contribution of long-range interactions.
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Affiliation(s)
- RC Ball
- Department of Physics, University of Warwick, CV4 7AL Coventry, United Kingdom
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24
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Zheng GP, Li M. Short-time dynamics of an ising system on fractal structures. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:6253-6259. [PMID: 11101957 DOI: 10.1103/physreve.62.6253] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/28/2000] [Indexed: 05/23/2023]
Abstract
The short-time critical relaxation of an Ising model on a Sierpinski carpet is investigated using Monte Carlo simulation. We find that when the system is quenched from high temperature to the critical temperature, the evolution of the order parameter and its persistence probability, the susceptibility, and the autocorrelation function all show power-law scaling behavior at the short-time regime. The results suggest that the spatial heterogeneity and the fractal nature of the underlying structure do not influence the scaling behavior of the short-time critical dynamics. The critical temperature, dynamic exponent z, and other equilibrium critical exponents beta and nu of the fractal spin system are determined accurately using conventional Monte Carlo simulation algorithms. The mechanism for short-time dynamic scaling is discussed.
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Affiliation(s)
- GP Zheng
- Department of Materials Science and Engineering, The Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218, USA
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25
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Gao Z, Yang ZR. Dynamic behavior of the Ziff-Gulari-Barshad model on fractal lattices: the influence of the order of ramification. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:2741-4. [PMID: 11970078 DOI: 10.1103/physreve.60.2741] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/05/1999] [Indexed: 04/18/2023]
Abstract
A catalytic reaction model, the Ziff-Gulari-Barshad model, is studied on fractal lattices, and the influence of the order of ramification of the lattice on the dynamic behavior of the model is investigated. According to the Monte Carlo simulation results, the order of ramification of the lattice is not crucial to the existence of the continuous transition. This is different from the equilibrium phase transitions in discrete-symmetry spin models (such as the Ising model). Our results indicate that the criterion of the existence of the reactive phase may be complicated.
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Affiliation(s)
- Z Gao
- Department of Physics and Institute of Theoretical Physics, Beijing Normal University, Beijing 100875, People's Republic of China
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26
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27
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Taguchi Y. Noninteger-dimensional hyper-Euclidean lattices on Sierpinski carpets. JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL 1999. [DOI: 10.1088/0305-4470/21/3/043] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/11/2022]
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Abstract
This study explores the use of fractal analysis in the numerical description of chromatin appearance in breast cytology. Images of nuclei from fine-needle aspiration biopsies of the breast are characterized in terms of their Minkowski and spectral fractal dimensions, for 19 patients with benign epithelial cell lesions and 22 with invasive ductal carcinomas. Chromatin appearance in breast epithelial cell nuclear images is demonstrated to be fractal, suggesting that the three-dimensional chromatin structure in these cells also has fractal properties. A statistically significant difference between the mean spectral dimensions of the benign and malignant cases is demonstrated. The two fractal dimensions are very weakly correlated. A statistically significant difference between the benign and malignant cases in lacunarity, a fractal property characterizing the size of holes or gaps in a texture, is found over a wide range of scales. These differences are particularly pronounced at the smallest and largest scales, corresponding respectively to fine-scale texture, indicating whether chromatin is clumped or fine, and to large-scale structures like nucleoli. Logistic regression and artificial neural network classification models are developed to classify unknown cases on the basis of fractal measures of chromatin texture. Using leave-one-out cross-validation, the best logistic regression classifier correctly diagnoses 95.1 per cent of the cases. The best neural network model can correctly classify all of the cases, but it is unclear whether this is due to overtraining. Fractal dimensions and lacunarity are useful tools for the quantitative characterization of chromatin appearance, and can potentially be incorporated into image analysis devices to assure the quality and reproducibility of diagnosis by breast fine-needle aspiration biopsy.
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Affiliation(s)
- A J Einstein
- Department of Biomathematical Sciences, Mount Sinai School of Medicine, New York, NY 10029, USA.
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38
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Abstract
Let
F
n
be the
n
th stage in the construction of the Sierpiński carpet. Let
R
n
be the electrical resistance of
F
n
when the left and right sides are each short-circuited, and a voltage is applied between them. We prove that there exists a constant
ρ
such that ¼
ρ
n
≼
R
n
≼ 4
ρ
n
. The motivation for this result came from the problem of establishing (
a
) the existence and (
b
) the value of the ‘spectral dimension’ of the Sierpiński carpet. In this and a subsequent paper, we settle (
a
) and give bounds for (
b
).
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Jezewski W. Critical behavior in a quasifractal Ising model. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:6029-6034. [PMID: 9965818 DOI: 10.1103/physreve.54.6029] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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40
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Perreau M, Peiro J, Berthier S. Percolation in random-Sierpin-acuteski carpets: A real space renormalization group approach. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 54:4590-4595. [PMID: 9965634 DOI: 10.1103/physreve.54.4590] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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41
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Tomczak P, Ferchmin AR, Richter J. Ground state of an antiferromagnetic Heisenberg spin system on a nontranslational lattice of dimension between one and two. PHYSICAL REVIEW. B, CONDENSED MATTER 1996; 54:395-401. [PMID: 9984272 DOI: 10.1103/physrevb.54.395] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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42
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Riera R. High-temperature series expansions for Ising-like systems on fractals. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:2579-2587. [PMID: 9961518 DOI: 10.1103/physreve.49.2579] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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43
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Yang ZR. Solvable Ising model on Sierpínski carpets: The partition function. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:2457-2460. [PMID: 9961490 DOI: 10.1103/physreve.49.2457] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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44
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Monceau PJ, Levy JC. Monodimensional effects on elastic and vibrational properties of lacunary networks. PHYSICAL REVIEW. B, CONDENSED MATTER 1994; 49:1026-1038. [PMID: 10010407 DOI: 10.1103/physrevb.49.1026] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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45
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Shi Y, Gong C. Critical dimensionalities of phase transitions on fractals. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1994; 49:99-103. [PMID: 9961194 DOI: 10.1103/physreve.49.99] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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46
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Blumenfeld R, Ball RC. Probe for morphology and hierarchical correlations in scale-invariant structures. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1993; 47:2298-2302. [PMID: 9960257 DOI: 10.1103/physreve.47.2298] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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47
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Potts model on infinitely ramified Sierpinski-gasket-type fractals and algebraic order at antiferromagnetic phases. PHYSICAL REVIEW. B, CONDENSED MATTER 1992; 46:11642-11656. [PMID: 10003053 DOI: 10.1103/physrevb.46.11642] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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48
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Ilgenfritz E, Schiller A, Markum H. Exploring the critical behavior of U(1) gauge theory on regular and fractal lattices by a finite-size analysis. PHYSICAL REVIEW. D, PARTICLES AND FIELDS 1992; 45:2949-2956. [PMID: 10014689 DOI: 10.1103/physrevd.45.2949] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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49
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Allain C, Cloitre M. Characterizing the lacunarity of random and deterministic fractal sets. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1991; 44:3552-3558. [PMID: 9906372 DOI: 10.1103/physreva.44.3552] [Citation(s) in RCA: 134] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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50
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Yung Q, Yang ZR. Crossover from a fractal lattice to a Euclidean lattice for the thermodynamic properties of a triplet-interaction Ising model. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 43:13342-13347. [PMID: 9997164 DOI: 10.1103/physrevb.43.13342] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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