Articles de revues : « Power-law scaling »

1

Tatlıer, M. « Power-law scaling behavior of membranes ». Journal of Membrane Science 182, n^o 1-2 (15 février 2001) : 183–93. http://dx.doi.org/10.1016/s0376-7388(00)00565-2.

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BURROUGHS, STEPHEN M., et SARAH F. TEBBENS. « UPPER-TRUNCATED POWER LAW DISTRIBUTIONS ». Fractals 09, n^o 02 (juin 2001) : 209–22. http://dx.doi.org/10.1142/s0218348x01000658.

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Power law cumulative number-size distributions are widely used to describe the scaling properties of data sets and to establish scale invariance. We derive the relationships between the scaling exponents of non-cumulative and cumulative number-size distributions for linearly binned and logarithmically binned data. Cumulative number-size distributions for data sets of many natural phenomena exhibit a "fall-off" from a power law at the largest object sizes. Previous work has often either ignored the fall-off region or described this region with a different function. We demonstrate that when a data set is abruptly truncated at large object size, fall-off from a power law is expected for the cumulative distribution. Functions to describe this fall-off are derived for both linearly and logarithmically binned data. These functions lead to a generalized function, the upper-truncated power law, that is independent of binning method. Fitting the upper-truncated power law to a cumulative number-size distribution determines the parameters of the power law, thus providing the scaling exponent of the data. Unlike previous approaches that employ alternate functions to describe the fall-off region, an upper-truncated power law describes the data set, including the fall-off, with a single function.

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Ton, Robert, et Andreas Daffertshofer. « Model selection for identifying power-law scaling ». NeuroImage 136 (août 2016) : 215–26. http://dx.doi.org/10.1016/j.neuroimage.2016.01.008.

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CAMPOS, PAULO R. A., VIVIANE M. DE OLIVEIRA et LEONARDO P. MAIA. « EMERGENCE OF ALLOMETRIC SCALING IN GENEALOGICAL TREES ». Advances in Complex Systems 07, n^o 01 (mars 2004) : 39–46. http://dx.doi.org/10.1142/s0219525904000044.

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We investigate the emergence of power-law scalings in genealogical trees. Especially, we study the topological properties of genealogical trees both in the neutral evolution and the selective evolution. In all instances, we observe that the topologies of these trees are well described by a power-law scaling [Formula: see text], where Ak is the number of nodes which are direct or indirect descendants of node k and Ck=∑jAj where the sum is taken over all nodes that contribute to Ak. This relation is well known in transportation networks as well as in metabolic networks, and it is referred to as allometric scaling. Furthermore, we observe a slight dependence of the scaling exponent η on the intensity of selection.

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Chen, Bo, Chunying Ma, Witold F. Krajewski, Pei Wang et Feipeng Ren. « Logarithmic transformation and peak-discharge power-law analysis ». Hydrology Research 51, n^o 1 (2 décembre 2019) : 65–76. http://dx.doi.org/10.2166/nh.2019.108.

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Abstract The peak-discharge and drainage area power-law relation has been widely used in regional flood frequency analysis for more than a century. The coefficients and can be obtained by nonlinear or log-log linear regression. To illustrate the deficiencies of applying log-transformation in peak-discharge power-law analyses, we studied 52 peak-discharge events observed in the Iowa River Basin in the United States from 2002 to 2013. The results show that: (1) the estimated scaling exponents by the two methods are remarkably different; (2) for more than 80% of the cases, the power-law relationships obtained by log-log linear regression produce larger prediction errors of peak discharge in the arithmetic scale than that predicted by nonlinear regression; and (3) logarithmic transformation often fails to stabilize residuals in the arithmetic domain, it assigns higher weight to data points representing smaller peak discharges and drainage areas, and it alters the visual appearance of the scatter in the data. The notable discrepancies in the scaling parameters estimated by the two methods and the undesirable consequences of logarithmic transformation raise caution. When conducting peak-discharge scaling analysis, especially for prediction purposes, applying nonlinear regression on the arithmetic scale to estimate the scaling parameters is a better alternative.

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Luo, Liang, et Lei-Han Tang. « Sub-diffusive scaling with power-law trapping times ». Chinese Physics B 23, n^o 7 (juillet 2014) : 070514. http://dx.doi.org/10.1088/1674-1056/23/7/070514.

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Bhattacharyya, Gautam, Anindya Datta, Swarup Kumar Majee et Amitava Raychaudhuri. « Power law scaling in universal extra dimension scenarios ». Nuclear Physics B 760, n^o 1-2 (janvier 2007) : 117–27. http://dx.doi.org/10.1016/j.nuclphysb.2006.10.018.

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Gupta, Hari M., et José R. Campanha. « Firms growth dynamics, competition and power-law scaling ». Physica A : Statistical Mechanics and its Applications 323 (mai 2003) : 626–34. http://dx.doi.org/10.1016/s0378-4371(03)00017-7.

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Kitzes, Justin. « Evidence for power‐law scaling in species aggregation ». Ecography 42, n^o 6 (8 février 2019) : 1224–25. http://dx.doi.org/10.1111/ecog.04159.

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Ferree, Thomas C., et Rudolph C. Hwa. « Power-law scaling in human EEG : relation to Fourier power spectrum ». Neurocomputing 52-54 (juin 2003) : 755–61. http://dx.doi.org/10.1016/s0925-2312(02)00760-9.

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Keeling, Matt, et Bryan Grenfell. « Stochastic dynamics and a power law for measles variability ». Philosophical Transactions of the Royal Society of London. Series B : Biological Sciences 354, n^o 1384 (29 avril 1999) : 769–76. http://dx.doi.org/10.1098/rstb.1999.0429.

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Since the discovery of a power law scaling between the mean and variance of natural populations, this phenomenon has been observed for a variety of species. Here, we show that the same form of power law scaling also occurs in measles case reports in England and Wales. Remarkably this power law holds over four orders of magnitude. We consider how the natural experiment of vaccination affects the slope of the power law. By examining simple generic models, we are able to predict the effects of stochasticity and coupling and we propose a new phenomenon associated with the critical community size.

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Brissaud, Quentin, et Victor C. Tsai. « Validation of a fast semi-analytic method for surface-wave propagation in layered media ». Geophysical Journal International 219, n^o 2 (5 septembre 2019) : 1405–20. http://dx.doi.org/10.1093/gji/ggz351.

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SUMMARY Green’s functions provide an efficient way to model surface-wave propagation and estimate physical quantities for near-surface processes. Several surface-wave Green’s function approximations (far-field, no mode conversions and no higher mode surface waves) have been employed for numerous applications such as estimating sediment flux in rivers, determining the properties of landslides, identifying the seismic signature of debris flows or to study seismic noise through cross-correlations. Based on those approximations, simple empirical scalings exist to derive phase velocities and amplitudes for pure power-law velocity structures providing an exact relationship between the velocity model and the Green’s functions. However, no quantitative estimates of the accuracy of these simple scalings have been reported for impulsive sources in complex velocity structures. In this paper, we address this gap by comparing the theoretical predictions to high-order numerical solutions for the vertical component of the wavefield. The Green’s functions computation shows that attenuation-induced dispersion of phase and group velocity plays an important role and should be carefully taken into account to correctly describe how surface-wave amplitudes decay with distance. The comparisons confirm the general reliability of the semi-analytic model for power-law and realistic shear velocity structures to describe fundamental-mode Rayleigh waves in terms of characteristic frequencies, amplitudes and envelopes. At short distances from the source, and for large near-surface velocity gradients or high Q values, the low-frequency energy can be dominated by higher mode surface waves that can be captured by introducing additional higher mode Rayleigh-wave power-law scalings. We also find that the energy spectral density for realistic shear-velocity models close to piecewise power-law models can be accurately modelled using the same non-dimensional scalings. The frequency range of validity of each power-law scaling can be derived from the corresponding phase velocities. Finally, highly discontinuous near-surface velocity profiles can also be approximated by a combination of power-law scalings. Analytical Green’s functions derived from the non-dimensionalization provide a good estimate of the amplitude and variations of the energy distribution, although the predictions are quite poor around the frequency bounds of each power-law scaling.

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Barenblatt, G. I. « Scaling laws for fully developed turbulent shear flows. Part 1. Basic hypotheses and analysis ». Journal of Fluid Mechanics 248 (mars 1993) : 513–20. http://dx.doi.org/10.1017/s0022112093000874.

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The present work consists of two parts. Here in Part 1, a scaling law (incomplete similarity with respect to local Reynolds number based on distance from the wall) is proposed for the mean velocity distribution in developed turbulent shear flow. The proposed scaling law involves a special dependence of the power exponent and multiplicative factor on the flow Reynolds number. It emerges that the universal logarithmic law is closely related to the envelope of a family of power-type curves, each corresponding to a fixed Reynolds number. A skin-friction law, corresponding to the proposed scaling law for the mean velocity distribution, is derived.In Part 2 (Barenblatt & Prostokishin 1993), both the scaling law for the velocity distribution and the corresponding friction law are compared with experimental data.

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Farrell-Gray, Catherine C., et Nicholas J. Gotelli. « ALLOMETRIC EXPONENTS SUPPORT A 3/4-POWER SCALING LAW ». Ecology 86, n^o 8 (août 2005) : 2083–87. http://dx.doi.org/10.1890/04-1618.

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Baccelli, François, et Anup Biswas. « On Scaling Limits of Power Law Shot-Noise Fields ». Stochastic Models 31, n^o 2 (3 avril 2015) : 187–207. http://dx.doi.org/10.1080/15326349.2014.990980.

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Lam, Chi-Hang, et Leonard M. Sander. « Exact scaling in surface growth with power-law noise ». Physical Review E 48, n^o 2 (1 août 1993) : 979–87. http://dx.doi.org/10.1103/physreve.48.979.

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Miller, Kai J., Larry B. Sorensen, Jeffrey G. Ojemann et Marcel den Nijs. « Power-Law Scaling in the Brain Surface Electric Potential ». PLoS Computational Biology 5, n^o 12 (18 décembre 2009) : e1000609. http://dx.doi.org/10.1371/journal.pcbi.1000609.

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Takahashi, Seiki, et Satoru Kobayashi. « Scaling power-law relations in asymmetrical minor hysteresis loops ». Journal of Applied Physics 107, n^o 6 (15 mars 2010) : 063903. http://dx.doi.org/10.1063/1.3357215.

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Heskestad, Gunnar. « Scaling the initial convective flow of power law fires ». Fire Safety Journal 42, n^o 3 (avril 2007) : 240–42. http://dx.doi.org/10.1016/j.firesaf.2006.12.004.

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Pilkington, Mark, et John P. Todoeschuck. « Power-law scaling behavior of crustal density and gravity ». Geophysical Research Letters 31, n^o 9 (7 mai 2004) : n/a. http://dx.doi.org/10.1029/2004gl019883.

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Baldassarri, Andrea, et Bernard Sapoval. « Power law statistics of cliff failures, scaling and percolation ». Earth Surface Processes and Landforms 40, n^o 8 (24 février 2015) : 1116–28. http://dx.doi.org/10.1002/esp.3703.

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Pudney, M. A., C. M. Carr, S. J. Schwartz et S. I. Howarth. « Near equipment magnetic field verification and scaling ». Geoscientific Instrumentation, Methods and Data Systems Discussions 3, n^o 2 (26 juillet 2013) : 437–58. http://dx.doi.org/10.5194/gid-3-437-2013.

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Abstract. Magnetic field measurements are essential to the success of many scientific space missions. Outside of the Earth's magnetic field the biggest potential source of magnetic field contamination of these measurements is emitted by the spacecraft. Spacecraft magnetic cleanliness is enforced through the application of strict ground verification requirements for spacecraft equipment and instruments. Due to increasingly strict AC magnetic field requirements, many spacecraft units cannot be verified on the ground using existing techniques. These measurements must instead be taken close to the equipment under test (EUT) and then extrapolated. A traditional dipole power law of −3 (with a field fall-off proportional to r−3) cannot be applied at these close distances without risk of underestimating the field emitted by the EUT, but we demonstrate that a power law of −2 is too conservative. We propose a compromise that uses a power law of −2 up to a distance equal to 3 times the unit size, beyond which a dipole power law can be applied. When extrapolating from a distance of 0.20 to 1.00 m from the centre of a 0.20 m wide EUT, we demonstrate that this method avoids an underprediction of the field, and is at least twice as accurate as performing the extrapolation with a fixed power law of −2.

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Zhuleku, J., J. Warnecke et H. Peter. « Stellar coronal X-ray emission and surface magnetic flux ». Astronomy & ; Astrophysics 640 (août 2020) : A119. http://dx.doi.org/10.1051/0004-6361/202038022.

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Context. Observations show that the coronal X-ray emission of the Sun and other stars depends on the surface magnetic field. Aims. Using power-law scaling relations between different physical parameters, we aim to build an analytical model to connect the observed X-ray emission to the surface magnetic flux. Methods. The basis for our model are the scaling laws of Rosner, Tucker & Vaiana (RTV) that connect the temperature and pressure of a coronal loop to its length and energy input. To estimate the energy flux into the upper atmosphere, we used scalings derived for different heating mechanisms, such as field-line braiding or Alfvén wave heating. We supplemented this with observed relations between active region size and magnetic flux and derived scalings of how X-ray emissivity depends on temperature. Results. Based on our analytical model, we find a power-law dependence of the X-ray emission on the magnetic flux, LX ∝ Φm, with a power-law index m being in the range from about one to two. This finding is consistent with a wide range of observations, from individual features on the Sun, such as bright points or active regions, to stars of different types and varying levels of activity. The power-law index m depends on the choice of the heating mechanism, and our results slightly favor the braiding and nanoflare scenarios over Alfvén wave heating. In addition, the choice of instrument will have an impact on the power-law index m because of the sensitivity of the observed wavelength region to the temperature of the coronal plasma. Conclusions. Overall, our simple analytical model based on the RTV scaling laws gives a good representation of the observed X-ray emission. Therefore we might be able to understand stellar coronal activity though a collection of basic building blocks, like loops, which we can study in spatially resolved detail on the Sun.

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Siena, M., A. Guadagnini, M. Riva et S. P. Neuman. « Extended power-law scaling of air permeabilities measured on a block of tuff ». Hydrology and Earth System Sciences Discussions 8, n^o 4 (15 août 2011) : 7805–43. http://dx.doi.org/10.5194/hessd-8-7805-2011.

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Abstract. We use three methods to identify power law scaling of (natural) log air permeability data collected by Tidwell and Wilson (1999) on the faces of a laboratory-scale block of Topopah Spring tuff: method of moments (M), extended power-law scaling also known as Extended Self-Similarity (ESS) and a generalized version thereof (G-ESS). All three methods focus on qth-order sample structure functions of absolute increments. Most such functions exhibit power-law scaling at best over a limited midrange of experimental separation scales, or lags, which are sometimes difficult to identify unambiguously by means of M. ESS and G-ESS extend this range in a way that renders power-law scaling easier to characterize. Most analyses of this type published to date concern time series or one-dimensional transects of spatial data associated with a unique measurement (support) scale. We consider log air permeability data having diverse support scales on the faces of a cube. Our analysis confirms the superiority of ESS and G-ESS over M in identifying the scaling exponents ξ(q) of corresponding structure functions of orders q, suggesting further that ESS is more reliable than G-ESS. The exponents vary in a nonlinear fashion with q as is typical of real or apparent (Guadagnini and Neuman, 2011; Guadagnini et al., 2011) multifractals. Our estimates of the Hurst scaling coefficient increase with support scale, implying a reduction in roughness (anti-persistence) of the log permeability field with measurement volume. ESS and G-ESS ratios between scaling exponents ξ(q) associated with various orders q show no distinct dependence on support volume or on two out of three Cartesian directions (there being no distinct power law scaling in the third direction). The finding by Tidwell and Wilson (1999) that log permeabilities associated with all tip sizes can be characterized by stationary variogram models, coupled with our findings that log permeability increments associated with the smallest tip size are approximately Gaussian and those associated with all tip size scales show nonlinear (multifractal) variations in ξ(q) with q, are consistent with a view of these data as a sample from a truncated version (tfBm) of self-affine fractional Brownian motion (fBm). Since in theory the scaling exponents, ξ(q), of tfBm vary linearly with q we conclude, in accord with Neuman (2010a, b, 2011), that nonlinear scaling in our case is not an indication of multifractality but an artifact of sampling from tfBm. This allows us to explain theoretically how power law scaling is extended by ESS. It further allows us to identify the functional form and estimate all parameters of the corresponding tfBm based on sample structure functions of first and second orders. Our estimate of lower cutoff is consistent with a theoretical support scale of the data.

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Telesca, L., et M. Lovallo. « Investigating non-uniform scaling behaviour in temporal fluctuations of seismicity ». Natural Hazards and Earth System Sciences 8, n^o 5 (1 septembre 2008) : 973–76. http://dx.doi.org/10.5194/nhess-8-973-2008.

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Abstract. Scaling behaviour in nonstationary time series can be successfully detected using the detrended fluctuation analysis (DFA). Observational time series often do not show a stable and uniform scaling behaviour, given by the presence of a unique clear scaling region. The deviations from uniform power-law scaling, which suggest the presence of changing dynamics in the system under study, can be identified and quantified using an appropriate instability index. In this framework, the scaling behaviour of the 1981–2007 seismicity in Umbria-Marche (central Italy), which is one of the most seismically active areas in Italy, was investigated. Significant deviations from uniform power-law scaling in the seismic temporal fluctuations were revealed mostly linked with the occurrence of rather large earthquakes or seismic clusters.

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CHEN, YANGUANG. « POWER-LAW DISTRIBUTIONS BASED ON EXPONENTIAL DISTRIBUTIONS : LATENT SCALING, SPURIOUS ZIPF'S LAW, AND FRACTAL RABBITS ». Fractals 23, n^o 02 (28 mai 2015) : 1550009. http://dx.doi.org/10.1142/s0218348x15500097.

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The difference between the inverse power function and the negative exponential function is significant. The former suggests a complex distribution, while the latter indicates a simple distribution. However, the association of the power-law distribution with the exponential distribution has been seldom researched. This paper is devoted to exploring the relationships between exponential laws and power laws from the angle of view of urban geography. Using mathematical derivation and numerical experiments, I reveal that a power-law distribution can be created through a semi-moving average process of an exponential distribution. For the distributions defined in a one-dimension space (e.g. Zipf's law), the power exponent is 1; while for those defined in a two-dimension space (e.g. Clark's law), the power exponent is 2. The findings of this study are as follows. First, the exponential distributions suggest a hidden scaling, but the scaling exponents suggest a Euclidean dimension. Second, special power-law distributions can be derived from exponential distributions, but they differ from the typical power-law distributions. Third, it is the real power-law distributions that can be related with fractal dimension. This study discloses an inherent link between simplicity and complexity. In practice, maybe the result presented in this paper can be employed to distinguish the real power laws from spurious power laws (e.g. the fake Zipf distribution).

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Tebbens, Sarah F., et Stephen M. Burroughs. « Self-Similar Criticality ». Fractals 11, n^o 03 (septembre 2003) : 221–31. http://dx.doi.org/10.1142/s0218348x03002117.

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Cumulative frequency-size distributions associated with many natural phenomena follow a power law. Self-organized criticality (SOC) models have been used to model characteristics associated with these natural systems. As originally proposed, SOC models generate event frequency-size distributions that follow a power law with a single scaling exponent. Natural systems often exhibit power law frequency-size distributions with a range of scaling exponents. We modify the forest fire SOC model to produce a range of scaling exponents. In our model, uniform energy (material) input produces events initiated on a self-similar distribution of critical grid cells. An event occurs when material is added to a critical cell, causing that material and all material in occupied non-diagonal adjacent cells to leave the grid. The scaling exponent of the resulting cumulative frequency-size distribution depends on the fractal dimension of the critical cells. Since events occur on a self-similar distribution of critical cells, we call this model Self-Similar Criticality (SSC). The SSC model may provide a link between fractal geometry in nature and observed power law frequency-size distributions for many natural systems.

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Siena, M., A. Guadagnini, M. Riva et S. P. Neuman. « Extended power-law scaling of air permeabilities measured on a block of tuff ». Hydrology and Earth System Sciences 16, n^o 1 (4 janvier 2012) : 29–42. http://dx.doi.org/10.5194/hess-16-29-2012.

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Abstract. We use three methods to identify power-law scaling of multi-scale log air permeability data collected by Tidwell and Wilson on the faces of a laboratory-scale block of Topopah Spring tuff: method of moments (M), Extended Self-Similarity (ESS) and a generalized version thereof (G-ESS). All three methods focus on q-th-order sample structure functions of absolute increments. Most such functions exhibit power-law scaling at best over a limited midrange of experimental separation scales, or lags, which are sometimes difficult to identify unambiguously by means of M. ESS and G-ESS extend this range in a way that renders power-law scaling easier to characterize. Our analysis confirms the superiority of ESS and G-ESS over M in identifying the scaling exponents, ξ(q), of corresponding structure functions of orders q, suggesting further that ESS is more reliable than G-ESS. The exponents vary in a nonlinear fashion with q as is typical of real or apparent multifractals. Our estimates of the Hurst scaling coefficient increase with support scale, implying a reduction in roughness (anti-persistence) of the log permeability field with measurement volume. The finding by Tidwell and Wilson that log permeabilities associated with all tip sizes can be characterized by stationary variogram models, coupled with our findings that log permeability increments associated with the smallest tip size are approximately Gaussian and those associated with all tip sizes scale show nonlinear variations in ξ(q) with q, are consistent with a view of these data as a sample from a truncated version (tfBm) of self-affine fractional Brownian motion (fBm). Since in theory the scaling exponents, ξ(q), of tfBm vary linearly with q we conclude that nonlinear scaling in our case is not an indication of multifractality but an artifact of sampling from tfBm. This allows us to explain theoretically how power-law scaling of our data, as well as of non-Gaussian heavy-tailed signals subordinated to tfBm, are extended by ESS. It further allows us to identify the functional form and estimate all parameters of the corresponding tfBm based on sample structure functions of first and second orders.

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Qiu, L. Y., H. Y. Liang, Y. B. Yang, H. X. Yang, T. Tian, Y. Xu et L. M. Duan. « Observation of generalized Kibble-Zurek mechanism across a first-order quantum phase transition in a spinor condensate ». Science Advances 6, n^o 21 (mai 2020) : eaba7292. http://dx.doi.org/10.1126/sciadv.aba7292.

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The Kibble-Zurek mechanism provides a unified theory to describe the universal scaling laws in the dynamics when a system is driven through a second-order quantum phase transition. However, for first-order quantum phase transitions, the Kibble-Zurek mechanism is usually not applicable. Here, we experimentally demonstrate and theoretically analyze a power-law scaling in the dynamics of a spin-1 condensate across a first-order quantum phase transition when a system is slowly driven from a polar phase to an antiferromagnetic phase. We show that this power-law scaling can be described by a generalized Kibble-Zurek mechanism. Furthermore, by experimentally measuring the spin population, we show the power-law scaling of the temporal onset of spin excitations with respect to the quench rate, which agrees well with our numerical simulation results. Our results open the door for further exploring the generalized Kibble-Zurek mechanism to understand the dynamics across first-order quantum phase transitions.

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Guđmundsson, Agust, et Nahid Mohajeri. « Relations between the scaling exponents, entropies, and energies of fracture networks ». Bulletin de la Société Géologique de France 184, n^o 4-5 (1 juillet 2013) : 373–82. http://dx.doi.org/10.2113/gssgfbull.184.4-5.373.

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Abstract Fracture networks commonly show power-law length distributions. Thermodynamic principles form the basis for understanding fracture initiation and growth, but have not been easily related to the power-law size distributions. Here we present the power-law scaling exponents and the calculated entropies of fracture networks from the Holocene part of the plate boundary in Iceland. The total number of tension fractures and normal faults used in these calculations is 565 and they range in length by five orders of magnitude. Each network can be divided into populations based on ‘breaks’ (abrupt changes) in the scaling exponents. The breaks, we suggest, are related to the comparatively long and deep fractures changing from tension fractures into normal faults and penetrating the contacts between the Holocene lava flows and the underlying and mechanically different Quaternary rocks. The results show a strong linear correlation (r = 0.84) between the population scaling exponents and entropies. The correlation is partly explained by the entropy (and the scaling exponent) varying positively with the arithmetic average and the length range (the difference between the longest and the shortest fracture) of the populations in each network. We show that similar scaling laws apply to other lineaments, such as streets. We propose that the power-law size distributions of fractures are a consequence of energy requirements for fracture growth.

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LI, XIAO-XIA, et JI-HUAN HE. « ALONG THE EVOLUTION PROCESS KLEIBER'S 3/4 LAW MAKES WAY FOR RUBNER'S SURFACE LAW : A FRACTAL APPROACH ». Fractals 27, n^o 02 (mars 2019) : 1950015. http://dx.doi.org/10.1142/s0218348x19500154.

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Rubner 1880 surface law reveals that the basal metabolic rate scales with body mass raised to the power of 2/3, which is geometrically correct and biologically relevant. However, Kleiber 1932 scaling law experimentally found that the scaling index was 3/4 instead of 2/3. There is no theory that can explain the Kleiber's data, explanations in Science in 1997 and later in Nature in 2002 for 3/4 scaling law for all life were apparently wrong. Here we show that Rubner's surface law was approximately correct, and it requires modification due to the fact that a cell is porous. Using fractal theory, the scaling index is about 0.7, 0.73, and 0.83, respectively, for inactive, active and motion statuses, and Kleiber's exponent can be fully explained by Rubner's law.

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Riedel, K. S. « Dimensionality correct power law scaling expressions for L-mode confinement ». Nuclear Fusion 31, n^o 5 (1 mai 1991) : 927–41. http://dx.doi.org/10.1088/0029-5515/31/5/009.

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Marrett, Randall, Orlando J. Ortega et Celinda M. Kelsey. « Extent of power-law scaling for natural fractures in rock ». Geology 27, n^o 9 (1999) : 799. http://dx.doi.org/10.1130/0091-7613(1999)027<0799:eoplsf>2.3.co;2.

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O’Malley, Daniel, John H. Cushman et Graham Johnson. « Scaling laws for fractional Brownian motion with power-law clock ». Journal of Statistical Mechanics : Theory and Experiment 2011, n^o 01 (14 janvier 2011) : L01001. http://dx.doi.org/10.1088/1742-5468/2011/01/l01001.

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Viswanathan, Gandhimohan M., C. K. Peng, H. Eugene Stanley et Ary L. Goldberger. « Deviations from uniform power law scaling in nonstationary time series ». Physical Review E 55, n^o 1 (1 janvier 1997) : 845–49. http://dx.doi.org/10.1103/physreve.55.845.

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Amar, Jacques G., et Fereydoon Family. « Crossover scaling in surface growth with truncated power-law noise ». Journal de Physique I 1, n^o 2 (février 1991) : 175–79. http://dx.doi.org/10.1051/jp1:1991123.

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Ashkenazy, Yosef, Shlomo Havlin, Plamen Ch Ivanov, Chung-K. Peng, Verena Schulte-Frohlinde et H. Eugene Stanley. « Magnitude and sign scaling in power-law correlated time series ». Physica A : Statistical Mechanics and its Applications 323 (mai 2003) : 19–41. http://dx.doi.org/10.1016/s0378-4371(03)00008-6.

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Collins, S. « Aspects of determining ÆBs : scaling and power-law divergences ». Nuclear Physics B - Proceedings Supplements 83-84, n^o 1-3 (mars 2000) : 271–73. http://dx.doi.org/10.1016/s0920-5632(00)00249-8.

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Pichler, Ch, R. Traxl et R. Lackner. « Power-law scaling of thermal conductivity of highly porous ceramics ». Journal of the European Ceramic Society 35, n^o 6 (juin 2015) : 1933–41. http://dx.doi.org/10.1016/j.jeurceramsoc.2014.12.004.

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Cohen, Joel E. « Species-abundance distributions and Taylor’s power law of fluctuation scaling ». Theoretical Ecology 13, n^o 4 (4 juillet 2020) : 607–14. http://dx.doi.org/10.1007/s12080-020-00470-x.

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Barlow, John, Michael Lim, Nick Rosser, David Petley, Matthew Brain, Emma Norman et Melanie Geer. « Modeling cliff erosion using negative power law scaling of rockfalls ». Geomorphology 139-140 (février 2012) : 416–24. http://dx.doi.org/10.1016/j.geomorph.2011.11.006.

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Kitzbichler, Manfred G., et Edward T. Bullmore. « Power Law Scaling in Human and Empty Room MEG Recordings ». PLOS Computational Biology 11, n^o 5 (8 mai 2015) : e1004175. http://dx.doi.org/10.1371/journal.pcbi.1004175.

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Wu, Songbai, Li Chen, Ninglian Wang, Shaohui Xu, Vincenzo Bagarello et Vito Ferro. « Variable power‐law scaling of hillslope Hortonian rainfall–runoff processes ». Hydrological Processes 33, n^o 22 (26 juillet 2019) : 2926–38. http://dx.doi.org/10.1002/hyp.13538.

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Le, Phong V. V., et Praveen Kumar. « Power law scaling of topographic depressions and their hydrologic connectivity ». Geophysical Research Letters 41, n^o 5 (6 mars 2014) : 1553–59. http://dx.doi.org/10.1002/2013gl059114.

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Burroughs, Stephen M., et Sarah F. Tebbens. « Power-law Scaling and Probabilistic Forecasting of Tsunami Runup Heights ». Pure and Applied Geophysics 162, n^o 2 (février 2005) : 331–42. http://dx.doi.org/10.1007/s00024-004-2603-5.

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Zingales, Massimiliano. « An exact thermodynamical model of power-law temperature time scaling ». Annals of Physics 365 (février 2016) : 24–37. http://dx.doi.org/10.1016/j.aop.2015.08.014.

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Sivakumar, C., et R. Francis. « On a Slightly Different Power Law-Scaling for the Flat Universe ». Journal of Scientific Research 12, n^o 4 (1 septembre 2020) : 569–74. http://dx.doi.org/10.3329/jsr.v12i4.46721.

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A slightly different power law-scaling fits to the picture of our 13.7 billion years old flat universe which is expanding presently at 67 km/s/Mpc with an acceleration. The model which is an attempt to retain power-law scaling in the light of the accepted facts about the universe we are living in, has a constant effective equation of state parameter as the cosmic fluid is a solution of matter, radiation and dark energy. It is successful in explaining the acceleration of universe which the normal power law fails if the present Hubble parameter is 67 km/s/Mpc and age of the universe is 13.7 billion years, and it is free from the defect of singularity.

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Barenblatt, G. I., A. J. Chorin et V. M. Prostokishin. « Scaling Laws for Fully Developed Turbulent Flow in Pipes ». Applied Mechanics Reviews 50, n^o 7 (1 juillet 1997) : 413–29. http://dx.doi.org/10.1115/1.3101726.

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Mathematical and experimental evidence is presented to the effect that the velocity profile in the intermediate region of turbulent shear flow in a pipe obeys a Reynolds-number dependent scaling (power) law rather than the widely believed von Ka´rma´n-Prandtl universal logarithmic law. In particular, it is shown that similarity theory and the Izakson-Millikan-von Mises overlap argument support the scaling law at least as much as they support the logarithmic law, while the experimental evidence overwhelmingly supports the scaling law. This review article includes 39 references.

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Varotsos, C. A., S. Lovejoy, N. V. Sarlis, M. N. Efstathiou et C. G. Tzanis. « On the scaling of the solar incident flux ». Atmospheric Chemistry and Physics Discussions 15, n^o 7 (15 avril 2015) : 10971–86. http://dx.doi.org/10.5194/acpd-15-10971-2015.

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Abstract. It was recently found that spectral solar incident flux (SIF) as a function of the ultraviolet wavelengths exhibits 1/f-type power-law correlations. In this study, an attempt was made to explore the SIF intrinsic dynamics vs. a wider range of wavelengths, from 115.5 to 629.5 nm. It seemed that the intermittency of SIF data set was very high and the revealed DFA-n exponents were close to unity thus again indicating 1/f power-law correlations. Moreover, the power spectral density was fitted algebraically with exponents close to unity. Eliminating the fitting of Planck formula at the Sun's effective temperature from SIF data set, scaling exponents very close to unity were derived, indicating that the 1/f scaling dynamics concern not the Planck's law but its fluctuations.

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Pudney, M. A., C. M. Carr, S. J. Schwartz et S. I. Howarth. « Near-magnetic-field scaling for verification of spacecraft equipment ». Geoscientific Instrumentation, Methods and Data Systems 2, n^o 2 (14 novembre 2013) : 249–55. http://dx.doi.org/10.5194/gi-2-249-2013.

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Abstract. Magnetic-field measurements are essential to the success of many scientific space missions. Outside of the earth's magnetic field the biggest potential source of magnetic-field contamination of these measurements is emitted by the spacecraft. Spacecraft magnetic cleanliness is enforced through the application of strict ground verification requirements for spacecraft equipment and instruments. Due to increasingly strict AC magnetic-field requirements, many spacecraft units cannot be verified on the ground using existing techniques. These measurements must instead be taken close to the equipment under test (EUT) and then extrapolated. A traditional dipole power law of −3 (with a field fall-off proportional to r−3) cannot be applied at these close distances without risk of underestimating the field emitted by the EUT, but we demonstrate that a power law of −2 is too conservative. We propose a compromise that uses a power law of −2 up to a distance equal to 3 times the unit size, beyond which a dipole power law can be applied. When extrapolating from a distance of 0.20 m to 1.00 m from the centre of a 0.20 m wide EUT, we demonstrate that this method avoids an under prediction of the field, and is at least twice as accurate as performing the extrapolation with a fixed power law of −2.

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