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Based on a sample of 7696 nonrecurrent and 566 recurrent sunspot groups, we have investigated the Gnevyshev–Waldmeier rule using the Catalogue of Solar Activity by R.S. Gnevysheva (1954–1978). The maximum recorded sunspot group lifetime is 160 days. The general Gnevyshev–Waldmeier rule for all lifetimes is Amax = (12.1 ± 0.70)LT (Amax is the maximum sunspot group area during the lifetime and LT is the sunspot group lifetime). However, the general rule has turned out to deviate from the derived linear form for the population of small short-living sunspot groups (SSG): it has a significant nonlinear scaling form Amax = (8.02 ± 0.41)LT(1.105±0.022). For the population of large long-living sunspot groups (LLG) there is a linear form Amax = (12.9 ± 1.1)LT. Allowance only for recurrent sunspot groups with lifetimes more than 40 days gives Amax = (13.93 ± 0.41)LT.

Accurately predicting energy requirements form a critical component for initializing dynamic mathematical models of metabolism. The majority of such existing estimates rely on linear regression models that predict total daily energy expenditure (TDEE) from age, gender, height, and body mass, however, there is evidence these predictors obey a power function. Baseline, free-living TDEE measured by doubly labeled water (DLW) in 20 studies with no overlapping subjects were obtained from the core lab at the University of Chicago and the University of Wisconsin-Madison (N = 2501 adults, 628 males, 1873 females). Linear regression models of log-transformed equations of the form: $${\rm{TDEE}} = \alpha _1M^{\beta _1}H^{\beta _2}$$ TDEE = α 1 M β 1 H β 2 and $${\rm{TDEE}} = {\rm{ln}}\left( {\alpha _1M^{\beta _1 + \gamma _1{\rm{Sex}}}H^{\beta _2 + \gamma _2{\rm{Sex}}}} \right)$$ TDEE = ln α 1 M β 1 + γ 1 Sex H β 2 + γ 2 Sex were developed to determine the values of the exponents of body mass (M (kg)) and height (H (cm)) along with a gender effect (Sex). A nonlinear curve fit was performed to develop a power model that also includes age $${\rm{TDEE}} = \alpha _1M^{\beta _1}H^{\beta _2} + \alpha _2{\mathrm{Age}}$$ TDEE = α 1 M β 1 H β 2 + α 2 Age . The power for body mass, β1 = 0.45 and the power for height was β2 = 1.52 in the database with both genders combined. Adding gender reduced these to β1 = 0.43 and β2 = 1.04. All terms were significant (p

The surface eigenstress model was further developed to study the surface-induced size-dependent ultimate biaxial tensile strength of thin films. First-principles calculations and molecular dynamics (MD) simulations were conducted on Au and Si thin films with different thicknesses. The atomistic calculations show that the ultimate biaxial tensile strength of Au thin films increases when the film thickness decreases, a typical the thinner the stronger phenomenon, while the Si thin films exhibit the thinner the weaker behavior. Nevertheless, both behaviors were perfectly predicted by a nonlinear scaling law developed from the surface eigenstress model. Moreover, the first-principles calculations put insight into the surface induced strengthening or weakening mechanism.

International audience; It has been known for a long time that the kinetics of diffusion-limited reactions can be quantified by the time needed for a diffusing molecule to reach a target: the first-passage time (FPT). So far the general determination of the mean first-passage time to a target in confinement has left aside aging media, such as glassy materials, cellular media, or cold atoms in optical lattices. Here we consider general non-Markovian scale-invariant diffusion processes, which model a broad class of transport processes of molecules in aging media, and demonstrate that all the moments of the FPT obey universal scalings with the confining volume with nontrivial exponents. Our analysis shows that a nonlinear scaling with the volume of the mean FPT, which quantities the mean reaction time, is the hallmark of aging and provides a general tool to quantify its impact on reaction kinetics in confinement.

Crime is a major threat to society's well-being but lacks a statistical characterization that could lead to uncovering some of its underlying mechanisms. Evidence of nonlinear scaling of urban indicators in cities, such as wages and serious crime, has motivated the understanding of cities as complex systems-a perspective that offers insights into resources limits and sustainability, but that usually neglects details of the indicators themselves. Notably, since the nineteenth century, criminal activities have been known to occur unevenly within a city; crime concentrates in such way that most of the offenses take place in few regions of the city. Though confirmed by different studies, this concentration lacks broad analyses on its characteristics, which hinders not only the comprehension of crime dynamics but also the proposal of sounding counter-measures. Here, we developed a framework to characterize crime concentration which divides cities into regions with the same population size. We used disaggregated criminal data from 25 locations in the U.S. and the U.K., spanning from 2 to 15 years of longitudinal data. Our results confirmed that crime concentrates regardless of city and revealed that the level of concentration does not scale with city size. We found that the distribution of crime in a city can be approximated by a power-law distribution with exponent α that depends on the type of crime. In particular, our results showed that thefts tend to concentrate more than robberies, and robberies more than burglaries. Though criminal activities present regularities of concentration, we found that criminal ranks have the tendency to change continuously over time-features that support the perspective of crime as a complex system and demand analyses and evolving urban policies covering the city as a whole.

Scaling methods are widely used in railgun technology. But nonlinear problems are seldom studied because of the complexity and uncertainty. To employ the scaling method better, this paper brings the nonlinear drag model to the solid armature railgun circuit. Several drag models are compared including friction drag model (FDM), atmosphere drag model (ADM), and current drag model (CDM). Then, nonlinear scaling relationships of these drag models are discussed. Furthermore, a novel experiment is constructed to compare the drag models at low velocity. At last, the nonlinear scaling method is verified by an example at high velocity. FDM is accurate enough to simulate a railgun system at any condition. Coefficient $\varepsilon$ in FDM should be used considering different test conditions. Scaling relationships of FDM could not be derived at low velocity because of nonlinearity. Although at high velocity, FDM could be simplified to CDM and realize linear scaling method (LSM). Further research indicates that the FDM could use the scaling method directly and realize approximate LSM. CDM is workable which satisfies the conditions of LSM in nature. Due to the work range of the solid armature, usually under 2.5 km/s, nonlinear scaling method could be used in a railgun with FDM. For a solid armature railgun, the $f_{\rm ADM}$ is too small compared with $f_{\rm FDM}$ and it could be ignored in analysis of a railgun, especially at low velocity. What should be noted is that the velocity scaling factor must be controlled above about one-quarter to avoid the nonlinear field.

We investigate the asymptotic behavior of solutions to semi-classical Schroedinger equations with nonlinearities of Hartree type. For a weakly nonlinear scaling, we show the validity of an asymptotic superposition principle for slowly modulated highly oscillatory pulses. The result is based on a high-frequency averaging effect due to the nonlocal nature of the Hartree potential, which inhibits the creation of new resonant waves. In the proof we make use of the framework of Wiener algebras., Comment: 13 pages; Version 2: Added Remark 2.2

An offset nonlinear scaling is developed for the ELMy H-mode by analysing the ITER H-mode database ITERH.DB2. The offset part of the stored energy is determined by the MHD stability of ELM, and the incremental confinement time of a nonlinear function of heating power is determined by the gyro-Bohm-like transport of the core plasma. This scaling predicts a lower confinement time for ITER than that predicted by general power-law scaling.

Scaling method has been used widely in railgun technology. But nonlinear problem is seldom studied because of the complexity and uncertainty. In order to employ the scaling method better, the paper brings the nonlinear drag model to the solid armature railgun circuit.

We report on a systematic investigation of the critical fluctuations in the complex conductivity of epitaxially grown ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+\ensuremath{\delta}}$ films for $T\ensuremath{\lesssim}{T}_{c}$ using a two-coil inductive technique at zero applied field. We observe the static three-dimensional (3D) $\mathrm{XY}$ critical exponent in the superfluid density near ${T}_{c}.$ Linear scaling analysis close to the critical temperature yields a dynamic critical exponent of $z\ensuremath{\approx}2.0$ for small drive currents, but nonlinear effects are seen to be important. At ${T}_{c},$ a nonlinear scaling analysis also yields a 3D dynamic exponent of $z=2.0\ifmmode\pm\else\textpm\fi{}0.1.$

The thermodynamic behavior of a fluid near a vapor-liquid and, hence, asymmetric critical point is discussed within a general ``complete'' scaling theory incorporating pressure mixing in the nonlinear scaling fields as well as corrections to scaling. This theory allows for a Yang-Yang anomaly in which \mu_{\sigma}^{\prime\prime}(T), the second temperature derivative of the chemical potential along the phase boundary, diverges like the specific heat when T\to T_{\scriptsize c}; it also generates a leading singular term, |t|^{2\beta}, in the coexistence curve diameter, where t\equiv (T-T_{\scriptsize c}) /T_{\scriptsize c}. The behavior of various special loci, such as the critical isochore, the critical isotherm, the k-inflection loci, on which \chi^{(k)}\equiv \chi(\rho,T)/\rho^{k} (with \chi = \rho^{2} k_{\scriptsize B}TK_{T}) and C_{V}^{(k)}\equiv C_{V}(\rho,T)/\rho^{k} are maximal at fixed T, is carefully elucidated. These results are useful for analyzing simulations and experiments, since particular, nonuniversal values of k specify loci that approach the critical density most rapidly and reflect the pressure-mixing coefficient. Concrete illustrations are presented for the hard-core square-well fluid and for the restricted primitive model electrolyte. For comparison, a discussion of the classical (or Landau) theory is presented briefly and various interesting loci are determined explicitly and illustrated quantitatively for a van der Waals fluid., Comment: 21 pages in two-column format including 8 figures

We report on calculations of the translational and rotational short-time self-diffusion coefficients $D^t_s$ and $D^r_s$ for suspensions of charge-stabilized colloidal spheres. These diffusion coefficients are affected by electrostatic forces and many-body hydrodynamic interactions (HI). Our computations account for both two-body and three-body HI. For strongly charged particles, we predict interesting nonlinear scaling relations $D^t_s\propto 1-a_t\phi^{4/3}$ and $D^r_s\propto 1-a_r\phi^2$ depending on volume fraction $\phi$, with essentially charge-independent parameters $a_t$ and $a_r$. These scaling relations are strikingly different from the corresponding results for hard spheres. Our numerical results can be explained using a model of effective hard spheres. Moreover, we perceptibly improve the known result for $D^t_s$ of hard sphere suspensions., Comment: 8 pages, LaTeX, 3 Postscript figures included using epsf

We study the diffusion of polymers through quenched short-range correlated random media by renormalization group (RG) methods, which allow us to derive universal predictions in the limit of long chains and weak disorder. We take local quenched random potentials with second momentv and the excluded-volume interactionu of the chain segments into account. We show that our model contains the relevant features of polymer diffusion in random media in the RG sense if we focus on the local entropic effects rather than on the topological constraints of a quenched random medium. The dynamic generating functional and the general structure of its perturbation expansion inu andv are derived. The distribution functions for the center-of-mass motion and the internal modes of one chain and for the correlation of the center of mass motions of two chains are calculated to one-loop order. The results allow for sufficient cross-checks to have trust in the one-loop renormalizability of the model. The general structure as well as the one-loop results of the integrated RG flow of the parameters are discussed. Universal results can be found for the effective static interactionw≔u−v≥0 and for small effective disorder coupling\(\bar v(l)\) on the intermediate length scalel. As a first physical prediction from our analysis, we determine the general nonlinear scaling form of the chain diffusion constant and evaluate it explicitly as Open image in new window for\(\bar v(l) \ll 1\).

The mean-square radius of gyration 〈${\mathit{R}}_{\mathit{G}}^{2}$〉 and a shape parameter \ensuremath{\Sigma}=〈${\mathit{R}}_{\mathit{G}\mathrm{min}}^{2}$〉/〈${\mathit{R}}_{\mathit{G}\mathrm{max}}^{2}$〉 are studied as a function of the number of bonds, bends, and branches of self-avoiding lattice trees on the square, triangular, and honeycomb lattices. We identify the universality classes, and exhibit the crossover scaling functions that connect them. We find (despite doubts recently raised) that there is a universal crossover from rods to self-avoiding walks, embodied in 〈${\mathit{R}}_{\mathit{G}}^{2}$〉\ensuremath{\sim}${\mathit{N}}^{2}$U(Nw), where w(z) is an appropriately chosen nonlinear scaling field reducing to the stiffness fugacity z as z\ensuremath{\rightarrow}0 ; that ``rigid trees'' (which are bond clusters that branch but do not bend) are in the same universality class as branched polymers or free trees; that the crossover from rods to rigid trees has the universal form 〈${\mathit{R}}_{\mathit{G}}^{2}$〉\ensuremath{\sim}${\mathit{N}}^{2}$W(${\mathit{Ny}}^{2}$), where y is the branching fugacity; and that the crossover from self-avoiding walks to branched polymers has the universal form 〈${\mathit{R}}_{\mathit{G}}^{2}$〉\ensuremath{\sim}${\mathit{N}}_{\mathit{s}}^{2\ensuremath{\nu}}$Y(${\mathit{Ny}}^{\mathrm{\ensuremath{\varphi}}}$), with ${\ensuremath{\nu}}_{\mathit{s}}$=3/4 and \ensuremath{\varphi}=55/32.

The ‘in-field’ magnetization of amorphous Fe 91 Zr 9 , Fe 90 Zr 10 and Fe 89 Co 1 Zr 10 alloys obeys the scaling equation of state in non-linear variables e = (T − T C )/T and h = H(T C /T) (T C = Curie point) from T ⋍ 0.4T C to T ⋍ 1.4T C while initial susceptibility follows the generalized Curie-Weiss law from T ⋍ T C to T ⋍ 1.4T C . Non-analytic corrections to the singular behaviour at T C dominate over the analytic ones in the critical region but the reverse is true for T ⪢ T C .

Thermodynamic functions in the vicinity of an ordinary critical point are expected to obey asymptotic scaling laws as $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and the ordering field, $h$, approach zero. However, the optimal scaling variables are the nonlinear scaling fields, ${g}_{t}=t+{b}_{t}{h}^{2}+{c}_{t}{t}^{2}+\ensuremath{\cdots}$ and ${g}_{h}=h(1+{c}_{h}t+\ensuremath{\cdots})$. The nonlinearities yield correction factors to the leading power-law (and scaling) variation of thermodynamic quantities, $L$, of the form ($1+{a}_{L}t+{b}_{L}{t}^{2}+\ensuremath{\cdots}$), etc., where the correction amplitudes ${a}_{L},{b}_{L},\dots{}$, are uniquely determined by the nonlinear scaling-field coefficients. It follows that "analytic" corrections to, e.g., the susceptibility, are directly related to those for the free energy and magnetization (in zero field). The term ${b}_{t}{h}^{2}$ also generates nonanalytic contributions such as an additive, energylike term, varying as ${|t|}^{1\ensuremath{-}\ensuremath{\alpha}}$ in the zero-field susceptibility, and factors like ($1+{c}_{L}{|h|}^{2\ensuremath{-}\frac{1}{\ensuremath{\Delta}}}$) on the critical isotherm, $t=0$. Irrelevant scaling fields yield further, in general distinct, nonanalytic corrections, and cause shifts in ${T}_{c}$ and various amplitudes although "universal" ratios remain constant.