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In this note, we revisit the scaling relations among “hatted critical exponents”, which were first derived by Ralph Kenna, Des Johnston, and Wolfhard Janke, and we propose an alternative derivation for some of them. For the scaling relation involving the behavior of the correlation function, we will propose an alternative form since we believe that the expression is erroneous in the work of Ralph and his collaborators.

We develop a methodology to compile an objective tremor catalog by utilizing distinctive event features that differentiate tectonic tremors from non‐tremor events, and combining the envelope cross‐correlation method with clustering technique and neural network. This approach enables tremor extraction without subjective criteria, allowing for the detection of previously overlooked short‐duration tremors. The event features employed to distinguish tremors and non‐tremor events are depth, the mean amplitudes at high and low frequencies, the ratio of these two amplitudes, and event duration. The duration is defined as the minimum period that contains 50% of the seismic energy. The application of this method to western Japan detects 1.7 times more tremors than the previous studies, with the durations of 0.3–∼100 s. The events with short durations are considered low‐frequency earthquakes. The relationship between seismic moment and duration of the detected tremors is consistent with the scaling law of slow earthquakes. Plain Language Summary Slow earthquakes are characterized by very slow underground deformation compared with regular (fast) earthquakes and are important for understanding the preparation period prior to large earthquakes. Tectonic tremors, which are a type of slow earthquakes, radiate tiny seismic waves with frequencies of several Hz, occur episodically and densely in space and time, and may last for long durations of up to several hundred seconds, which is much longer than the durations of fast earthquakes of equivalent magnitude. In this study, we detect and differentiate tectonic tremors from fast earthquakes and anthropogenic events. We do this using a set of event features, without relying on subjective criteria. The durations of the detected tremors range from 0.3 to ∼100 s, and they appear consistent with a previously proposed scaling relationship for slow earthquakes. This result suggests that fast earthquakes and slow earthquakes have different physical mechanisms. Key Points We compile a more complete tectonic tremor catalog for western Japan using a clustering method based on event features Event duration, newly defined using energy radiation, clearly separates tectonic tremors from fast earthquakes Tectonic tremors, ranging in duration from 0.3 to 100 s, are consistent with the scaling law of slow earthquakes

Aim Species–area relationships (SARs) are fundamental scaling laws in ecology although their shape is still disputed. At larger areas, power laws best represent SARs. Yet, it remains unclear whether SARs follow other shapes at finer spatial grains in continuous vegetation. We asked which function describes SARs best at small grains and explored how sampling methodology or the environment influence SAR shape. Location Palaearctic grasslands and other non‐forested habitats. Taxa Vascular plants, bryophytes and lichens. Methods We used the GrassPlot database, containing standardized vegetation‐plot data from vascular plants, bryophytes and lichens spanning a wide range of grassland types throughout the Palaearctic and including 2,057 nested‐plot series with at least seven grain sizes ranging from 1 cm2 to 1,024 m2. Using nonlinear regression, we assessed the appropriateness of different SAR functions (power, power quadratic, power breakpoint, logarithmic, Michaelis–Menten). Based on AICc, we tested whether the ranking of functions differed among taxonomic groups, methodological settings, biomes or vegetation types. Results The power function was the most suitable function across the studied taxonomic groups. The superiority of this function increased from lichens to bryophytes to vascular plants to all three taxonomic groups together. The sampling method was highly influential as rooted presence sampling decreased the performance of the power function. By contrast, biome and vegetation type had practically no influence on the superiority of the power law. Main conclusions We conclude that SARs of sessile organisms at smaller spatial grains are best approximated by a power function. This coincides with several other comprehensive studies of SARs at different grain sizes and for different taxa, thus supporting the general appropriateness of the power function for modelling species diversity over a wide range of grain sizes. The poor performance of the Michaelis–Menten function demonstrates that richness within plant communities generally does not approach any saturation, thus calling into question the concept of minimal area.

A prevailing paradigm suggests that species richness increases with area in a decelerating way. This ubiquitous power law scaling, the species–area relationship, has formed the foundation of many conservation strategies. In spatially complex ecosystems, however, the area may not be the sole dimension to scale biodiversity patterns because the scale-invariant complexity of fractal ecosystem structure may drive ecological dynamics in space. Here, we use theory and analysis of extensive fish community data from two distinct geographic regions to show that riverine biodiversity follows a robust scaling law along the two orthogonal dimensions of ecosystem size and complexity (i.e., the dual scaling law). In river networks, the recurrent merging of various tributaries forms fractal branching systems, where the prevalence of branching (ecosystem complexity) represents a macroscale control of the ecosystem’s habitat heterogeneity. In the meantime, ecosystem size dictates metacommunity size and total habitat diversity, two factors regulating biodiversity in nature. Our theory predicted that, regardless of simulated species’ traits, larger and more branched “complex” networks support greater species richness due to increased space and environmental heterogeneity. The relationships were linear on logarithmic axes, indicating power law scaling by ecosystem size and complexity. In support of this theoretical prediction, the power laws have consistently emerged in riverine fish communities across the study regions (Hokkaido Island in Japan and the midwestern United States) despite hosting different fauna with distinct evolutionary histories. The emergence of dual scaling law may be a pervasive property of branching networks with important implications for biodiversity conservation.

Hepatitis significantly increases the global disease burden and has become a major public health issue worldwide. China is a high-risk area for viral hepatitis, which is also a serious public health problem. The scaling relationship between various types of hepatitis and population size was explained by a scaling law. Fixed-effects and random-effects meta-analyses were used to calculate a combined index of β based on the single-scale index from 2004 to 2023. Furthermore, the X11 process was employed to identify the structural components of the time series of various types of hepatitis. In the past 20 years, the proportion of patients with viral hepatitis in Zhejiang Province has changed significantly, and hepatitis B remains the main type of hepatitis, accounting for approximately 70% of all hepatitis cases. The proportion of hepatitis C and E cases has been increasing, whereas the proportion of hepatitis A cases has been decreasing since 2004 and has remained at a low level (approximately 3%) since 2010. The combined scaling exponents of hepatitis A, hepatitis B, hepatitis C, hepatitis E and unclassified hepatitis based on the random effects model were 0.88 (95% confidence interval(CI): 0.78 to 0.98), 0.78 (95% CI: 0.70 to 0.86), 1.18 (95% CI: 1.11 to 1.26), 0.91 (95% CI: 0.86 to 0.97) and 0.89 (95% CI: 0.79 to 1.00), respectively. In the past 20 years, the epidemic situation of hepatitis A, hepatitis B and unclassified hepatitis has shown a significant downward trend, whereas the proportions of hepatitis C and hepatitis E among those with viral hepatitis have increased annually. The combined scaling exponent and development trends of the five types of hepatitis show significant heterogeneity. Overall, hepatitis C exhibits superlinear characteristics, whereas other types of hepatitis exhibit sublinear characteristics. Different types of hepatitis exhibit distinct epidemic characteristics and require targeted prevention and control measures.

Analyses of urban scaling laws assume that observations in different cities are independent of the existence of nearby cities. Here we introduce generative models and data-analysis methods that overcome this limitation by modelling explicitly the effect of interactions between individuals at different locations. Parameters that describe the scaling law and the spatial interactions are inferred from data simultaneously, allowing for rigorous (Bayesian) model comparison and overcoming the problem of defining the boundaries of urban regions. Results in five different datasets show that including spatial interactions typically leads to better models and a change in the exponent of the scaling law.

This paper addresses a non-traditional approach for the scaling-law fluid-flows described by fractal scaling-law vector calculus associated with the Mandelbrot scaling law. Their quantum equations were proposed to control the fluid-flows associated with the Mandelbrot scaling law. This gives a new insight into the descriptions for the scaling-law behaviors of the fluid-flows in the Mandelbrot scaling-law phenomena.

Topological defects—locations of local mismatch of order—are a universal concept playing important roles in diverse systems studied in physics and beyond, including the universe, various condensed matter systems, and recently, even life phenomena. Among these, liquid crystal has been a platform for studying topological defects via visualization, yet it has been a challenge to resolve three-dimensional structures of dynamically evolving singular topological defects. Here, we report a direct confocal observation of nematic liquid crystalline defect lines, called disclinations, relaxing from an electrically driven turbulent state. We focus in particular on reconnections, characteristic of such line defects. We find a scaling law for in-plane reconnection events, by which the distance between reconnecting disclinations decreases by the square root of time to the reconnection.Moreover, we show that apparently asymmetric dynamics of reconnecting disclinations is actually symmetric in a comoving frame, in marked contrast to the twodimensional counterpart whose asymmetry is established.We argue, with experimental supports, that this is because of energetically favorable symmetric twist configurations that disclinations take spontaneously, thanks to the topology that allows for rotation of the winding axis. Our work illustrates a general mechanism of such spontaneous symmetry restoring that may apply beyond liquid crystal, which can take place if topologically distinct asymmetric defects in lower dimensions become homeomorphic in higher dimensions and if the symmetric intermediate is energetically favorable.

In this study, we propose the general calculus operators based on the Richardson scaling law and Korcak scaling law. The Richardson-scaling-law calculus is considered to investigate the Fourier-like law for the scaling-law flow of the heat in the heat-transfer process. The Korcak-scaling-law calculus is used to model the Darcy-like law for describing the scaling-law flow of the fluid in porous medium. The formulas are as the special cases of the topology calculus proposed for descriptions of the fractal scaling-law behaviors in nature phenomena.