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Foamed concrete possesses characteristics such as high strength-to-weight ratio and low density, and widely used to reduce dead loads on the structure and foundation, contributes to energy conservation, and lowers the labor cost during construction. In this paper, the objective is to propose prediction relation for the compressive strength of foamed concrete by fractal theory. A theoretical relation was derived for the compressive strength relating to porosity based on the fractal model for foamed concrete. The proposed relation stands out compared to empirical model since it employs easily measurable parameter, the fractal dimension of porous structure in foamed concrete. The fractal dimension of porous structure can be calculated from the scaling law of the compressive strength of foamed concrete. The fractal model for porous structure serves as a simple and effective tool for predicting the compressive strength of foamed concrete because of its ease in application. The prediction relation of the compressive strength developed in this paper is found to match well with the measured strength.

The detrending moving average (DMA) algorithm is one of the best performing methods to quantify the long-term correlations in nonstationary time series. As many long-term correlated time series in real systems contain various trends, we investigate the effects of polynomial trends on the scaling behaviors and the performances of three widely used DMA methods including backward algorithm (BDMA), centered algorithm (CDMA) and forward algorithm (FDMA). We derive a general framework for polynomial trends and obtain analytical results for constant shifts and linear trends. We find that the behavior of the CDMA method is not influenced by constant shifts. In contrast, linear trends cause a crossover in the CDMA fluctuation functions. We also find that constant shifts and linear trends cause crossovers in the fluctuation functions obtained from the BDMA and FDMA methods. When a crossover exists, the scaling behavior at small scales comes from the intrinsic time series while that at large scales is dominated by the constant shifts or linear trends. We also derive analytically the expressions of crossover scales and show that the crossover scale depends on the strength of the polynomial trends, the Hurst index, and in some cases (linear trends for BDMA and FDMA) the length of the time series. In all cases, the BDMA and the FDMA behave almost the same under the influence of constant shifts or linear trends. Extensive numerical experiments confirm excellently the analytical derivations. We conclude that the CDMA method outperforms the BDMA and FDMA methods in the presence of polynomial trends.

The laws governing life should be as simple as possible; however, theoretical investigations into allometric laws have become increasingly complex, with the long-standing debate over the scaling exponent in allometric laws persisting. This paper re-examines the same biological phenomenon using two different scales. On a macroscopic scale, a cell surface appears smooth, but on a smaller scale, it exhibits a fractal-like porous structure. To elaborate, a few examples are given. Employing the two-scale fractal theory, we theoretically predict and experimentally verify the scaling exponent values for basal, active, and maximal metabolic rates. This paper concludes that Rubner’s 2/3 law and Kleiber’s 3/4 law are two facets of the same truth, manifested across different scale approximations.

We present the scaling law of the pinning force F_P in intermediate and high temperature superconductors, the factors which influence its field and temperature dependence, the limitations imposed by the enlarged anisotropy and high operating temperatures, giant creep included. The theoretical developments which incorporate the new features of these classes of superconductors, most of them based on the Anderson–Kim theory, prove to be a useful source of information relative to the nature of the pinning and the characteristic fields. The use of models based on thermal activation integrates into scaling the tail of pinning force and also substantiates the use of the irreversibility field H_irr as scale field. Finally, the data on scaling law in the two class of superconductors are presented and discussed.

We consider crack propagation in brittle nonlinear elastic materials in the context of quasi-static evolutions of energetic type. Given a sequence of self-similar domains nΩ on which the imposed boundary conditions scale according to Bažant's law, we show, in agreement with several experimental data, that the corresponding sequence of evolutions converges (for n → ∞) to the evolution of a crack in a brittle linear-elastic material.

Blast-induced traumatic brain injury (bTBI) has become a signature injury in recent military conflicts and terrorist attacks. However, the mechanisms and thresholds for such injury are still unknown. In this paper, effort has been made toward establishing the threshold due to primary blast based on the published injury data in the rat. Peak incident overpressure and pulse duration of the incident wave were used as predictors and the injury risk curves for the rat were derived via a linear logistic regression analysis. A scaling law based on body mass was then used to scale the tolerance curves from the rat to the pig and the human. The injury risk curve for bTBI was compared with that for the lung. The results reveal different injury mechanisms between these two organs. The developed injury curves can be used in the design of personal protective equipment against primary bTBI.

Tipping points have been actively studied in various applications as well as from a mathematical viewpoint. A main technique to theoretically understand early-warning signs for tipping points is to use the framework of fast–slow stochastic differential equations. A key assumption in many arguments for the existence of variance and auto-correlation growth before a tipping point is to use a linearization argument, i.e. the leading-order term governing the deterministic (or drift) part of a stochastic differential equation is linear. This assumption guarantees a local approximation via an Ornstein–Uhlenbeck process in the normally hyperbolic regime before, but sufficiently bounded away from, a bifurcation. In this paper, we generalize the situation to leading-order nonlinear terms for the setting of one fast variable. We work in the quasi-steady regime and prove that the fast variable has a well-defined stationary distribution and we calculate the scaling law for the variance as a bifurcation-induced tipping point is approached. We cross-validate the scaling law numerically. Furthermore, we provide a computational study for the predictability using early-warning signs for leading-order nonlinear terms based upon receiver-operator characteristic curves.

Urban traffic network is a typical complex system, in which movements of tremendous microscopic traffic participants (pedestrians, bicyclists and vehicles) form complicated spatial and temporal dynamics. We collected flow volumes data on the time-dependent activity of a typical urban traffic network, finding that the coupling between the average flux and the fluctuation on individual links obeys a certain scaling law, with a wide variety of scaling exponents between 1/2 and 1. These scaling phenomena can explain the interaction between the nodes' internal dynamics (i.e. queuing at intersections, car-following in driving) and changes in the external (network-wide) traffic demand (i.e. the every day increase of traffic amount during peak hours and shocking caused by traffic accidents), allowing us to further understand the mechanisms governing the transportation system's collective behavior. Multiscaling and hotspot features are observed in the traffic flow data as well. But the reason why the separated internal dynamics are comparable to the external dynamics in magnitude is still unclear and needs further investigations.

The excited states of some odd A nuclei near the stability line have been systematically investigated from light to intermediate mass with the spherical relativistic mean field (RMF) model. The ratio between the valence nucleon root-mean-square (RMS) radius and the core rms radius, the separation energy and the density distribution have been deduced as signatures for halo or skin structure. We have presented the scaling laws of the ratio of valence particle rms radii and square-potential radii versus the scaled separation energies. The probability for a valence particle being out of the binding potential has also been extracted. We proposed a relaxed necessary condition for nuclear halo occurrence.

The application of graph theory in structural biology offers an alternative means of studying 3D models of large macromolecules such as proteins. The radius of gyration, which scales with exponent $\sim 0.4$, provides quantitative information about the compactness of the protein structure. In this study, we combine two proven methods, the graph-theoretical and the fundamental scaling laws, to study 3D protein models. This study shows that the mean node degree (MND) of the protein graphs, which scales with exponent 0.038, is scale-invariant. In addition, proteins that differ in size have a highly similar node degree distribution. Linear regression analysis showed that the graph parameters (radius, diameter, and mean eccentricity) can explain up to 90% of the total radius of gyration variance. Thus, the graph parameters of radius, diameter, and mean eccentricity scale match along with the same exponent as the radius of gyration. The main advantage of graph eccentricity compared to the radius of gyration is that it can be used to analyze the distribution of the central and peripheral amino acids/nodes of the macromolecular structure.