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It is shown that if the initial condition of the Cauchy problem for the diffusion equation on a general discrete ultrametric space is spherically symmetric with respect to some point, then this problem has an exact analytical solution. An exact solution of this problem is presented for pure ultrametric diffusion, as well as for ultrametric diffusion with a reaction sink concentrated at the center of spherical symmetry. Conditions on the ultrametric and the distribution of the number of states in ultrametric spheres are found that lead at large times to the asymptotic behavior of the solutions obtained in the form of a power law modulated by a bounded function that is log-periodic under some additional conditions. [ABSTRACT FROM AUTHOR]

We consider a formal model of polytheistic religion, where worshippers choose gods to address, and expect benefits in return. At the game-theoretic equilibrium, they choose gods optimally depending on how the gods return favors, as a function of divine efficiency. This efficiency can be modeled based on a random growth model of prayer benefit, and results in worship acts following power laws. A reduction in the range of random events associated to the various divinities, or to their range of interpretations, leads to more concentration of worship towards less divinities. [ABSTRACT FROM AUTHOR]

The Takayasu aggregation model is a paradigmatic model of aggregation with mass injection, known to exhibit a power law distribution of mass over a range which grows in time. Working in one dimension we find that the mass profile in addition shows distinctive dynamic condensates which collectively hold a substantial portion of the mass (approximately 80 % when injection and diffusion rates are equal) and lead to a substantial hump in the scaled distribution. To track these, we monitor the largest mass within a growing coarsening length. An interesting outcome of extremal statistics is that the mean of the globally largest mass in a finite system grows as a power law in time, modulated by strong multiplicative logarithms in both time and system size. At very long times, in a finite system, the state consists of a power-law-distributed background with a condensate whose mass increases linearly with time. [ABSTRACT FROM AUTHOR]

Ultrasound B-scan images are traditionally formed from the envelope of the received radiofrequency echoes, but the image texture is dominated by granular speckle patterns. Longstanding efforts at speckle reduction and deconvolution have been developed to lessen the detrimental aspects of speckle. However, we now propose an alternative approach to estimation (and image rendering) of the underlying fine grain scattering density of tissues based on power law constraints. The key steps are a whitening of the spectrum of the received signal while conforming to the original envelope shape and statistics, followed by a power law filtering in accordance with the known scattering behavior of tissues. This multiple step approach results in a high-spatial-resolution map of scattering density that is constrained by the most important properties of scattering from tissues. Examples from in vivo liver scans are shown to illustrate the change in image properties from this framework.

Ultrasound B-scan images are traditionally formed from the envelope of the received radiofrequency echoes, but the image texture is dominated by granular speckle patterns. Longstanding efforts at speckle reduction and deconvolution have been developed to lessen the detrimental aspects of speckle. However, we now propose an alternative approach to estimation (and image rendering) of the underlying fine grain scattering density of tissues based on power law constraints. The key steps are a whitening of the spectrum of the received signal while conforming to the original envelope shape and statistics, followed by a power law filtering in accordance with the known scattering behavior of tissues. This multiple step approach results in a high-spatial-resolution map of scattering density that is constrained by the most important properties of scattering from tissues. Examples from in vivo liver scans are shown to illustrate the change in image properties from this framework. [ABSTRACT FROM AUTHOR]

In this work we suggest that the common cause for the development of various power laws is the existence of a suitable exchangeable quantity between the agents of a set. Examples of such exchangeable quantities, leading to eponymous power laws, include money (Pareto’s Law), scientific knowledge (Lotka’s Law), people (Auerbach’s Law), and written or verbal information (Zipf’s Law), as well as less common cases like bullets during deadly conflicts, recognition in social networks, heat between the atmosphere and sea-ice floes, and, finally, mass of water vapors between pores in solids. This last case is examined closely in the present article based on extensive experimental data. It is shown that the transferred mass between pores, which eventually grow towards a power law distribution, may be expressed using different parameters, either transferred surface area, or transferred volume, or transferred pore length or transferred pore anisotropy. These distinctions lead to different power laws of variable strength as reflected by the corresponding exponent. The exponents depend quantitatively on the spread of frequency distribution of the examined parameter and tend to zero as the spread of distribution tends to a single order of magnitude. A comparison between the energy and the entropy of different kinds of pore distributions reveals that these two statistical parameters are linearly related, implying that the system poise at a critical state and the exchangeable quantities are the most convenient operations helping to keep this balance.

Scholarship demonstrates that urban systems follow a power law population distribution if the population has full labor mobility. Theoretically, subpopulations should also follow a power law population distribution if that subpopulation also has full labor mobility. Examining city population distributions for White and Black Americans across US metropolitan areas from 1910 to 2020 shows that the White distribution mostly conforms to both Zipf's and Gibrat's laws throughout this period. In contrast, the Black population does not follow either law until the second half of the 20th century, a result that is consistent with theories of restricted mobility out of the South for Black Americans during the Jim Crow era. [ABSTRACT FROM AUTHOR]

In this work we suggest that the common cause for the development of various power laws is the existence of a suitable exchangeable quantity between the agents of a set. Examples of such exchangeable quantities, leading to eponymous power laws, include money (Pareto's Law), scientific knowledge (Lotka's Law), people (Auerbach's Law), and written or verbal information (Zipf's Law), as well as less common cases like bullets during deadly conflicts, recognition in social networks, heat between the atmosphere and sea-ice floes, and, finally, mass of water vapors between pores in solids. This last case is examined closely in the present article based on extensive experimental data. It is shown that the transferred mass between pores, which eventually grow towards a power law distribution, may be expressed using different parameters, either transferred surface area, or transferred volume, or transferred pore length or transferred pore anisotropy. These distinctions lead to different power laws of variable strength as reflected by the corresponding exponent. The exponents depend quantitatively on the spread of frequency distribution of the examined parameter and tend to zero as the spread of distribution tends to a single order of magnitude. A comparison between the energy and the entropy of different kinds of pore distributions reveals that these two statistical parameters are linearly related, implying that the system poise at a critical state and the exchangeable quantities are the most convenient operations helping to keep this balance. [ABSTRACT FROM AUTHOR]

We investigate the market value of rarity theoretically and empirically. Prior studies find that the market value of rarity follows a power law, but this finding lacks a theoretical foundation. We provide a micro-foundation for this finding, demonstrating that the observed power law emerges in a competitive market where agents have rank-dependent utility preferences. The model leads to two new theoretical insights: (i) the rank of an item within a set of close substitutes and the quantity of that item known to exist are both natural measures of rarity, but rank is predicted to perform better; (ii) there is a systematic relationship between the estimated slope and intercept from a regression of log price on log rank. When we test the model on data from over 4000 auction records of rare baseball cards, we find that a regression with only log rank and grading company explains 60% of the variation in log price. [ABSTRACT FROM AUTHOR]

Entropy can signify different things. For instance, heat transfer in thermodynamics or a measure of information in data analysis. Many entropies have been introduced, and it can be difficult to ascertain their respective importance and merits. Here, we consider entropy in an abstract sense, as a functional on a probability space, and we review how being able to handle the trivial case of non-interacting systems, together with the subtle requirement of extensivity, allows for a systematic classification of the functional form. [ABSTRACT FROM AUTHOR]

The notion of tail adversarial stability has been proven useful in obtaining limit theorems for tail dependent time series. Its implication and advantage over the classical strong mixing framework has been examined for max-linear processes, but not yet studied for additive linear processes. In this article, we fill this gap by verifying the tail adversarial stability condition for regularly varying additive linear processes. We in addition consider extensions of the result to a stochastic volatility generalization and to a max-linear counterpart. We also address the invariance of tail adversarial stability under monotone transforms. Some implications for limit theorems in statistical context are also discussed. [ABSTRACT FROM AUTHOR]

Isotopes have been widely applied in a variety of scientific subjects; many aspects of isotopes, however, remain not well understood. In this study, I investigate the relation between the number of neutrons (N) and the number of protons (Z) in stable isotopes of non-radioactive elements and long-lived isotopes of radioactive elements at the double-linear scale (conventional Segrè chart) and the double-logarithmic scale. Statistical analyses show that N is a power-law function of Z for these isotopes: N = 0.73 x Z[sup 116]. This power-law relation provides better predictions for the numbers of neutrons in stable isotopes of nonradioactive elements and long-lived isotopes of radioactive elements than the linear relation on the conventional Segrè chart. The power-law pattern reveled here offers empirical guidance for probing long-lived isotopes of unknown radioactive elements. [ABSTRACT FROM AUTHOR]

The science of complexity aims to answer the question of what rules nature chooses when assembling the basic constituents of matter and energy into structures and dynamical patterns that cascade through the entire hierarchy of scales in the Universe. A related phenomenon – natural language – can successfully mirror such structures as reflected by its ability to encode and transmit information about them and among them. It is thus legitimate to expect that natural language carries the essence of complexity. And indeed, in the human's speaking and writing it is particularly true that more is different. Natural language thus deserves a central place in the related quantitative study within the science of complexity. With this in mind the present review summarizes the main methodological concepts used in this domain and documents their applicability and utility in identifying universal as well as system-specific features of natural language in its written representation in several major Western languages. In particular, three main complexity-related current research trends in quantitative linguistics are exhaustively covered. The first part addresses the issue of word frequencies in texts and, in particular, demonstrates that taking punctuation into consideration largely restores scaling whose violation in the Zipf's law for the most frequent words is commonly modelled by the so-called Mandelbrot's correction. The second part introduces methods inspired by time series analysis, used in studying various kinds of long-range correlations in written texts. The related time series are generated on the basis of text partition into sentences or into phrases between consecutive punctuation marks. It turns out that these series develop features often found in signals generated by complex systems: the presence of long-range correlations along with fractal or even multifractal structures. Moreover, it appears that the distances between consecutive punctuation marks quite universally across languages comply with the discrete variant of the Weibull distribution, often appearing in survival analysis. In the third part, the application of the network formalism to natural language is reviewed, particularly in the context of word-adjacency networks whose structure reflects the word co-occurrence in texts. Various parameters characterizing topology of such networks can be used for classification of texts, for example, from a stylometric perspective. Network approach can also be applied in semantic analysis to represent a hierarchy of words and associations between them based on their meaning. Structure of such networks turns out to be significantly different from that observed in random networks, revealing genuine properties of language. Finally, punctuation appears to have a significant impact not only on the language's information-carrying ability but also on its key statistical properties, hence it seems recommended to consider punctuation marks on a par with words. [ABSTRACT FROM AUTHOR]

While well-established literature argues that realized variances are close to a lognormal distribution, this study follows Benoit Mandelbrot by taking a fractal perspective. Using power laws to model realized foreign exchange rate variances, our findings indicate that power laws offer an alternative to the lognormal in terms of goodness-of-fit tests. Further, our analysis shows that estimated power law exponents for seven out of nine realized FX variances are α ^ < 3 , which indicates that the variance of realized variance is statistically undefined. We conclude that the foreign exchange rate market is far riskier than earlier believed. By implication, documented research in an enormous body of literature that draws conclusions from variance analyses stands on shaky grounds. [ABSTRACT FROM AUTHOR]

Charged aerosol detection, increasingly recognized for quantifying pharmaceutical compounds with weak ultraviolet absorption, is a universal detection technique for high‐performance liquid chromatography (HPLC). Charged aerosol detection shows a non‐linear response with increasing analyte concentration over a wide range, limiting its versatility in various analytical applications. In this work, a co‐optimization strategy for power function value (PFV) and power laws was proposed and applied to broaden the linear range of the standard curve of saccharides in Qishen Yiqi dripping pills using the HPLC‐charged aerosol detection (HPLC‐CAD) method. Power function values for all analytes were optimized based on empirical models. Subsequently, the optimum power laws were investigated based on a preferred PFV. Additionally, various regression equations were evaluated to ensure the accuracy and precision of the results. With the optimized PFV and power law, the ordinary least squares model demonstrated a satisfactory fit. The optimal PFVs and power laws expanded the standard curve's linear range by 2.7 times compared to default settings, reducing model uncertainty. This paper presents a vital method for developing a multi‐component quantitative HPLC‐CAD approach without external data transformation outside the provided software, especially suitable for analytical applications of traditional Chinese medicine with significant quality differences. [ABSTRACT FROM AUTHOR]

Why do fractals appear in so many domains of science? What is the physical principle that generates them? While it is true that fractals naturally appear in many physical systems, it has so far been impossible to derive them from first physical principles. However, a proposed interpretation could shed light on the inherent principle behind the creation of fractals. This is the multiscale thermodynamic perspective, which states that an increase in external energy could initiate energy transport mechanisms that facilitate the dissipation or release of excess energy at different scales. Within this framework, it is revealed that power law patterns, and to a lesser extent, fractals, can emerge as a geometric manifestation to dissipate energy in response to external forces. In this context, the exponent of these power law patterns (thermodynamic fractal dimension D ) serves as an indicator of the balance between entropy production at small and large scales. Thus, when a system is more efficient at releasing excess energy at the microscopic (macroscopic) level, D tends to increase (decrease). While this principle, known as Principium luxuriæ, may sound promising for describing both multiscale and complex systems, there is still uncertainty about its true applicability. Thus, this work explores different physical, astrophysical, sociological, and biological systems to attempt to describe and interpret them through the lens of the Principium luxuriæ. The analyzed physical systems correspond to emergent behaviors, chaos theory, and turbulence. To a lesser extent, the cosmic evolution of the universe and geomorphology are examined. Biological systems such as the geometry of human organs, aging, human brain development and cognition, moral evolution, Natural Selection, and biological death are also analyzed. It is found that these systems can be reinterpreted and described through the thermodynamic fractal dimension. Therefore, it is proposed that the physical principle that could be behind the creation of fractals is the Principium luxuriæ, which can be defined as "Systems that interact with each other can trigger responses at multiple scales as a manner to dissipate the excess energy that comes from this interaction". That is why this framework has the potential to uncover new discoveries in various fields. For example, it is suggested that the reduction in D in the universe could generate emergent behavior and the proliferation of complexity in numerous fields or the reinterpretation of Natural Selection. [ABSTRACT FROM AUTHOR]

Abstract Cross‐affiliation emerges as a new and fast‐developing means to promote collaboration in financial research. We find that the average number of affiliations reported per author in the top‐three finance journals increases steadily from 1.1 to 1.3 from 1995 to 2016. Scale‐free power laws characterize the resulting highly‐skewed distributions of top finance journal publications of worldwide institutions. We propose an explanation of the scale‐invariance, based on a network model featuring nonlinear growth and linear preferential attachment. The model indicates that success‐breeds‐success engenders 87% of total publications and hence the dispersion in research output, while accelerated growth of collaboration reduces the heterogeneity.

In previous studies using eBird citizen data, bird abundance rankings followed a power law distribution. Our research delves into the “granular size” concept within these power laws, likening birds to firms. We identified 13 bird species as being the granular size, representing species with significant ecosystem impact, akin to major corporations in an economy. In particular, these species are urban, raising concerns about the eBird database’s sampling bias. Using the economic concept of granular size, we argue that the eBird database may be inherently unreliable.

Observations of power laws in neural activity data have raised the intriguing notion that brains may operate in a critical state. One example of this critical state is ‘avalanche criticality’, which has been observed in various systems, including cultured neurons, zebrafish, rodent cortex, and human EEG. More recently, power laws were also observed in neural populations in the mouse under an activity coarse-graining procedure, and they were explained as a consequence of the neural activity being coupled to multiple latent dynamical variables. An intriguing possibility is that avalanche criticality emerges due to a similar mechanism. Here, we determine the conditions under which latent dynamical variables give rise to avalanche criticality. We find that populations coupled to multiple latent variables produce critical behavior across a broader parameter range than those coupled to a single, quasi-static latent variable, but in both cases, avalanche criticality is observed without fine-tuning of model parameters. We identify two regimes of avalanches, both critical but differing in the amount of information carried about the latent variable. Our results suggest that avalanche criticality arises in neural systems in which activity is effectively modeled as a population driven by a few dynamical variables and these variables can be inferred from the population activity.

Simple Summary: Using eBird citizen science data, researchers identified power law patterns in bird abundance rankings. This study examines the concept of "granular size" related to these patterns, and determined 13 specific species to be the granular size. The granular size of the bird rank abundance distribution might have something to do with the number of species, which, like big businesses in an economy, have a disproportionately big effect on their environment. However, keep in mind that these species may be considered special simply because they were all discovered in cities. In previous studies using eBird citizen data, bird abundance rankings followed a power law distribution. Our research delves into the "granular size" concept within these power laws, likening birds to firms. We identified 13 bird species as being the granular size, representing species with significant ecosystem impact, akin to major corporations in an economy. In particular, these species are urban, raising concerns about the eBird database's sampling bias. Using the economic concept of granular size, we argue that the eBird database may be inherently unreliable. [ABSTRACT FROM AUTHOR]

Cross‐affiliation emerges as a new and fast‐developing means to promote collaboration in financial research. We find that the average number of affiliations reported per author in the top‐three finance journals increases steadily from 1.1 to 1.3 from 1995 to 2016. Scale‐free power laws characterize the resulting highly‐skewed distributions of top finance journal publications of worldwide institutions. We propose an explanation of the scale‐invariance, based on a network model featuring nonlinear growth and linear preferential attachment. The model indicates that success‐breeds‐success engenders 87% of total publications and hence the dispersion in research output, while accelerated growth of collaboration reduces the heterogeneity. [ABSTRACT FROM AUTHOR]

Non-classical characterizations of strange attractors assign occupancy times and numbers to each of the hypercubes that cover them, local information, and generate statistical distributions over such quantities, global information, at the expense of losing all the local ones. The confrontation between local and global information would allow statistical quantification, important for the characterization of the self-similarity of the involved attractor: correspondence between global and local properties are expected as manifestations of the self-similarity that characterizes that fractal set (statistical self-similarity). Hence, the importance of clarifying what the local distributions would be. The core of this work is the presentation of statistical distributions of the mentioned times and numbers, and quantities derived from them, linked to each hypercube of a set that corresponds to a small fraction of their total, that confirm the correspondence between local and global statistical properties. [ABSTRACT FROM AUTHOR]

Simple Summary: There are a lot of only a few kinds of birds. Scientists want to describe this fact with the right statistical distribution. Most of the time, the lognormal distribution is used. A new study, however, finds that the number of bird species is best described by a log left-skewed distribution. For the same set of data, we examine the rank abundance distribution rather than the species abundance distribution. In this case, the number of birds in each rank is based on a power law distribution. Only a few bird species are abundant. Understanding the abundance distribution of bird species is critical for conservation efforts because rare species may be more vulnerable to habitat loss, climate change, and other threats. According to new data, a log left-skewed distribution, rather than a lognormal distribution, better adjusts to the abundance distribution of bird species. We look at the rank abundance distribution rather than the species abundance distribution that use the same data and find three power laws: for the top four species; for the abundant species minus the top four; and for the rare species. [ABSTRACT FROM AUTHOR]

Entropy can signify different things. For instance, heat transfer in thermodynamics or a measure of information in data analysis. Many entropies have been introduced, and it can be difficult to ascertain their respective importance and merits. Here, we consider entropy in an abstract sense, as a functional on a probability space, and we review how being able to handle the trivial case of non-interacting systems, together with the subtle requirement of extensivity, allows for a systematic classification of the functional form.

Why do fractals appear in so many domains of science? What is the physical principle that generates them? While it is true that fractals naturally appear in many physical systems, it has so far been impossible to derive them from first physical principles. However, a proposed interpretation could shed light on the inherent principle behind the creation of fractals. This is the multiscale thermodynamic perspective, which states that an increase in external energy could initiate energy transport mechanisms that facilitate the dissipation or release of excess energy at different scales. Within this framework, it is revealed that power law patterns, and to a lesser extent, fractals, can emerge as a geometric manifestation to dissipate energy in response to external forces. In this context, the exponent of these power law patterns (thermodynamic fractal dimension D) serves as an indicator of the balance between entropy production at small and large scales. Thus, when a system is more efficient at releasing excess energy at the microscopic (macroscopic) level, D tends to increase (decrease). While this principle, known as Principium luxuriæ, may sound promising for describing both multiscale and complex systems, there is still uncertainty about its true applicability. Thus, this work explores different physical, astrophysical, sociological, and biological systems to attempt to describe and interpret them through the lens of the Principium luxuriæ. The analyzed physical systems correspond to emergent behaviors, chaos theory, and turbulence. To a lesser extent, the cosmic evolution of the universe and geomorphology are examined. Biological systems such as the geometry of human organs, aging, human brain development and cognition, moral evolution, Natural Selection, and biological death are also analyzed. It is found that these systems can be reinterpreted and described through the thermodynamic fractal dimension. Therefore, it is proposed that the physical principle that could be behind the creation of fractals is the Principium luxuriæ, which can be defined as “Systems that interact with each other can trigger responses at multiple scales as a manner to dissipate the excess energy that comes from this interaction”. That is why this framework has the potential to uncover new discoveries in various fields. For example, it is suggested that the reduction in D in the universe could generate emergent behavior and the proliferation of complexity in numerous fields or the reinterpretation of Natural Selection.

When modeling a directed social network, one choice is to use the traditional preferential attachment model, which generates power-law tail distributions. In traditional directed preferential attachment, every new edge is added sequentially into the network. However, real datasets often have only coarse timestamps, which means several new edges are created at the same timestamp. Previous analyses on the evolution of social networks reveal that after reaching a stable phase, the growth of edge counts in a network follows a non-homogeneous Poisson process with a constant rate across the day but varying rates from day to day. Taking such empirical observations into account, we propose a modified preferential attachment model with Poisson edge growth, and study its asymptotic behavior. This new model is then fitted to real datasets using an extreme value estimation approach. [ABSTRACT FROM AUTHOR]

We develop an equilibrium model for market impact of trades when investors with private signals execute via a trading desk. Fat tails in the signal distribution lead to a power law for price impact, while the impact is logarithmic for lighter tails. Moreover, the tail distribution of the equilibrium trade volume obeys a power law. The spread decreases with the degree of noise trading and increases with the number of insiders. In case of a monopolistic insider, the last slice traded against the limit order book is priced at the fundamental value of the asset reminiscent of [17]. However, competition among insiders leads to aggressive trading, hence vanishing profit in the limit. The model also predicts that the order book flattens as the amount of noise trading increases converging to a model with proportional transactions costs with non-vanishing spread. [ABSTRACT FROM AUTHOR]

We analyse features of the patterns formed from a simple model for a martensitic phase transition that fragments the unit square into rectangles. This is a fragmentation model that can be encoded by a general branching random walk. An important quantity is the distribution of the lengths of the interfaces in the pattern, and we establish limit theorems for some of the asymptotics of the interface profile. In particular, we are able to use a general branching process to show almost sure power law decay of the number of interfaces of at least a certain size and a general branching random walk to examine the numbers of rectangles of a certain aspect ratio. In doing so we extend a theorem on the growth of the general branching random walk as well as developing results on the tail behaviour of the limiting random variable in our general branching process. [ABSTRACT FROM AUTHOR]

Motivated by the complexity of network data, we propose a directed hybrid random network that mixes preferential attachment (PA) rules with uniform attachment rules. When a new edge is created, with probability p ∈ (0 , 1) , it follows the PA rule. Otherwise, this new edge is added between two uniformly chosen nodes. Such mixture makes the in- and out-degrees of a fixed node grow at a slower rate, compared to the pure PA case, thus leading to lighter distributional tails. For estimation and inference, we develop two numerical methods which are applied to both synthetic and real network data. We see that with extra flexibility given by the parameter p, the hybrid random network provides a better fit to real-world scenarios, where lighter tails from in- and out-degrees are observed. [ABSTRACT FROM AUTHOR]

We propose a new mechanism for generating power laws. Starting from a random walk, we first outline a simple derivation of the Fokker–Planck equation. By analogy, starting from a certain Markov chain, we derive a master equation for power laws that describes how the number of cascades changes over time (cascades are consecutive transitions that end when the initial state is reached). The partial differential equation has a closed form solution which gives an explicit dependence of the number of cascades on their size and on time. Furthermore, the power law solution has a natural cut-off, a feature often seen in empirical data. This is due to the finite size a cascade can have in a finite time horizon. The derivation of the equation provides a justification for an exponent equal to 2, which agrees well with several empirical distributions, including Richardson’s Law on the size and frequency of deadly conflicts. Nevertheless, the equation can be solved for any exponent value. In addition, we propose an urn model where the number of consecutive ball extractions follows a power law. In all cases, the power law is manifest over the entire range of cascade sizes, as shown through log–log plots in the frequency and rank distributions.

Urbanization is one of man's greatest challenges. Its handling requires a better understanding of orderliness in the demographic–socioeconomic–entrepreneurial domain of human settlements. Operating business enterprises are manifestations of successful entrepreneurship, which is the characteristic of interest here. Non-linear entrepreneurial proportionalities can be detected through the use of log–log regressions (power law analyses). Such analyses revealed many entrepreneurial proportionalities in datasets of a large number of U.S. counties. This enabled the examination of the temporal and geographic sensitivities of three entrepreneurial types: total entrepreneurship (expressed in total enterprise numbers), new entrepreneurship (the ability to successfully start enterprises of types not yet present), and existing entrepreneurship (the ability to start more enterprises of types already present). Stress testing of the entrepreneurial proportionalities during a period of economic growth (2000 to 2007) followed by a period of economic decline (the so-called Great Recession from 2007 to 2010) enabled the examination of a hypothesis that suggested that the entrepreneurial proportionalities are not temporally or geographically sensitive. The hypothesis is accepted for new and existing entrepreneurship. Total entrepreneurship is geographically sensitive, but not temporally. There is apparently no lack of entrepreneurship in human settlements. Their total entrepreneurship (expressed as total enterprise numbers) appears to be a function of their population sizes and prosperity/poverty levels. [ABSTRACT FROM AUTHOR]

This paper consists of a general consideration of a seismic system as a subsystem of another, larger system, exchanging with it by extensive dynamical quantities in a sequential push mode. It is shown that, unlike an isolated closed system described by the Liouville differential equation of the first order in time, it is described by a fractional differential equation of a distributed equation in the interval (0, 1] order. The key characteristic of its motion is a spectral function, representing the order distribution over the interval. As a specific case of the process, a system with single-point spectrum is investigated. It follows the fractional Poisson process method evolution, obeying via a time-fractional differential equation with a unique order. The article ends with description of statistical estimation of parameters of seismic shocks imitated by Monte Carlo simulated fractional Poisson process. [ABSTRACT FROM AUTHOR]

Random network models generated using sparse exchangeable graphs have provided a mechanism to study a wide variety of complex real-life networks. In particular, these models help with investigating power-law properties of degree distributions, number of edges, and other relevant network metrics which support the scale-free structure of networks. Previous work on such graphs imposes a marginal assumption of univariate regular variation (e.g., power-law tail) on the bivariate generating graphex function. In this paper, we study sparse exchangeable graphs generated by graphex functions which are multivariate regularly varying. We also focus on a different metric for our study: the distribution of the number of common vertices (connections) shared by a pair of vertices. The number being high for a fixed pair is an indicator of the original pair of vertices being connected. We find that the distribution of number of common connections are regularly varying as well, where the tail indices of regular variation are governed by the type of graphex function used. Our results are verified on simulated graphs by estimating the relevant tail index parameters. [ABSTRACT FROM AUTHOR]

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To assist with the detection of bubbles and negative bubbles in financial markets, a criterion is introduced to indicate whether a market is likely to be in a superexponential regime (where growth in such a regime would correspond to an asset price bubble and decline to an negative bubble) as opposed to "normal" exponential behavior typified by a constant rate of growth or decline. The criterion is founded on the Johansen–Ledoit–Sornette model of asset dynamics in a bubble and is derived from a linear fit to observed data with a nonlinear time transformation with parameters distributed uniformly in their permitted ranges. Making use of expected values rather than the underlying distribution, the criterion is straightforward and efficient to compute and can in principle be applied in real time to intra-day markets as well as longer timescales. In some circumstances, the criterion is shown to have certain predictive qualities when applied to a portfolio of stocks, and could be used as input into algorithmic trading strategies. A simple strategy is described which is based on market reversion predictions of a portfolio of stocks and which in back-testing generates notable returns. [ABSTRACT FROM AUTHOR]

This paper examines the spatial and temporal distribution of all COVID-19 cases from January to June 2020 against the underlying distribution of population in the United States. It is found that, as time passes, COVID-19 cases become a power law with cutoff, resembling the underlying spatial distribution of populations. The power law implies that many states and counties have a low number of cases, while only a few highly populated states and counties have a high number of cases. To further differentiate patterns between the underlying populations and COVID-19 cases, we derived their inherent hierarchy or spatial heterogeneity characterized by the ht-index. We found that the ht-index of COVID-19 cases persistently approaches that of the populations; that is, 5 and 7 at the state and county levels, respectively. Mapping the ht-index of COVID-19 cases against that of populations shows that the pandemic is largely shaped by the underlying population with the R-square value between infection and population up to 0.82.

The Rajeev–Ranken (RR) model is a Hamiltonian system describing screw-type nonlinear waves (screwons) of wavenumber k in a scalar field theory pseudodual to the 1+1D SU(2) principal chiral model. Classically, the RR model based on a quadratic Hamiltonian on a nilpotent/Euclidean Poisson algebra is Liouville integrable. Upon adopting canonical variables in a slightly extended phase space, the model was interpreted as a novel 3D cylindrically symmetric quartic oscillator with a rotational energy. Here, we examine the spectral statistics and dispersion relation of quantized screwons via numerical diagonalization validated by variational and perturbative approximations. We also derive a semiclassical estimate for the cumulative level distribution which compares favorably with the one from numerical diagonalization. The spectrum shows level crossings typical of an integrable system. The i th unfolded nearest neighbor spacings are found to follow Poisson statistics for small i. Nonoverlapping spacing ratios also indicate that successive spectral gaps are independently distributed. After displaying universal linear behavior over energy windows of short lengths, the spectral rigidity saturates at a length and value that scales with the square-root of energy. For strong coupling λ and intermediate k , we argue that reduced screwon energies can depend only on the product λ k. Numerically, we find power law dependences on λ and k with an approximately common exponent 2 / 3 provided the angular momentum quantum number l is small compared to the number of nodes n in the radial wavefunction. On the other hand, for the ground state n = l = 0 , the common exponent becomes 1. • Screwons are quantized nonlinear waves in a 1+1D scalar field theory. • Semiclassical screwon energy distribution agrees with numerical diagonalization. • Unfolded spacing and spacing ratio distributions follow from Poisson statistics. • Spectral rigidity has nonuniversal saturation and oscillations with power law scaling. • Screwon energies grow as 2/3-rds power of strong coupling and wavenumber. [ABSTRACT FROM AUTHOR]

Four different South African groups are principally involved with research about entrepreneurship, namely groups focused on: (i) entrepreneurship, (ii) small towns, (iii) local economic development, and (iv) microeconomics of towns. There is limited cooperation between these groups, which if overcome could contribute to the reduction of community poverty in towns. Two hypotheses are examined to sketch the importance of quantified entrepreneurial knowledge: 1. New entrepreneurship, i.e. the ability to successfully start businesses of types not yet present, is linked with entrepreneurial strength in the tradable sectors of towns. 2 Entrepreneurial strength in the tradable sector reduces poverty in towns. A range of power law (log–log) relationships reflect the entrepreneurial complexity of towns and enables tests of the hypotheses. The first is rejected and the second accepted. Strength in the tradable sector of towns reduces community poverty. Greater research cooperation should be sought. [ABSTRACT FROM AUTHOR]

Tropical precipitation clusters exhibit power‐law frequency distributions in area and volume (integrated precipitation), implying a lack of characteristic scale in tropical convective organization. However, it remains unknown what gives rise to the power laws and how the power‐law exponents for area and volume are related to one another. Here, we explore the perspective that precipitation clusters are islands above a convective threshold on a rough column‐water‐vapor (CWV) topography. This perspective is supported by the agreement between the precipitation clusters and CWV islands in their frequency distributions as well as fractal dimensions. Power laws exist for CWV islands at different thresholds through the CWV topography, suggesting that the existence of power laws is not specifically related to local precipitation dynamics, but is rather a general feature of CWV islands. Furthermore, the frequency distributions and fractal dimensions of the clusters can be reproduced when the CWV field is modeled to be self‐affine with a roughness exponent of 0.3. Self‐affine scaling theory relates the statistics of precipitation clusters to the roughness exponent; it also relates the power‐law slopes for area and volume without involving the roughness exponent. Thus, the perspective of precipitation clusters as CWV islands provides a useful framework to consider many statistical properties of these precipitation clusters, particularly given that CWV is well‐observed over a wide range of length‐scales in the Tropics. However, the statistics of CWV islands at the convective threshold imply a smaller roughness than is inferred from the power spectrum of the bulk CWV field, and further work is needed to understand the scaling of the CWV field. [ABSTRACT FROM AUTHOR]

Используя статистическую модель связанных событий и данные сейсмического каталога КФ ФИЦ ГС РАН за период 01.01.1962-31.12.2002 (n = 79283 события) для зоны субдукции Курило-Камчатской островной дуги (50°-60° с.ш., 156°-166° в.д., классы 4.1-16.1, глубины 0-600 км) построены последовательности событий, предшествующих главному событию заданной магнитуды и связанных с ним в пространственно-временной области. Проведён статистический анализ наличия степенных закономерностей в этих последовательностях на основании релаксационной модели деформационной активности. Степенные закономерности были выделены только в последовательностях связанных событий, предшествующих главному событию с энергетическим классом меньше 12 (показатель степени 0.5-0.7). Это объясняется большим объёмом выборки для главных событий такого класса и небольшой пространственно-временной областью их подготовки, что уменьшает вероятность попадания афтершоков от более крупных событий, предшествующих главному удару, в рассматриваемые последовательности.

Chinese Torreya (Torreya grandis cv. Merrillii) is an important economic tree in China, but there are limited studies on its seed production. We analyzed the patterns of historical seed production at two major sites (Zhaojiazhen and Jidongzhen) for Chinese Torreya from different perspectives. The results indicated that there were no 3-year or multiyear cycles in its seed production. A positive correlation existed between the average seed production and the average annual air temperature in 5 or 10 years at both study sites. There was no trend of the increasing coefficient of variance (cv) in seed production, but the cv generally increased before 1975, and became flat after that time. Frequency power law existed in seed production at both sites, but Taylor’s Law existed only at Zhaojiazhen. The multiscale entropy decreased with time scales, and the patterns were similar at both sites. Our research results provide a new understanding of seed production for Chinese Torreya.