Since the seminal work of Bak et al. [11], it has been tempting to try to connect such **power** laws with self-organized criticality (SOC). In existing models of SOC, such as sand- pile, sliding-block, forest-ﬁre, evolution, etc., a dynamic process plays a central role, yielding avalanches whose sizes follow a **power** **law** frequency distribution. Elucidating general mech- anisms underlying **power** **law** behaviors in biological and social sciences is a challenging task. We present here empirical evidence for a **power** **law** distribution in a new context; that of horse racing. We also propose a simple model to explain the phenomenon. We believe that the model is relevant beyond the speciﬁc context of horse racing, as it might explain how **power** **law** distributions emerge in social sciences, especially in a large assembly of interconnected human activities.

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A nonlinear Helmholtz equation for optical materials with regimes of **power**-**law** type of nonlinearity is proposed. This model captures broad beam evolution at any angle with respect to the reference direction in a wide range of media, including some semiconductors, doped glasses and liquid crystals. Novel exact analytical soliton solutions are presented for a generic nonlinearity, within which known Kerr solitons comprise a subset. Three new general conservation laws are also reported. Analysis and numerical simulations examine the stability of the Helmholtz **power**- **law** solitons. A new propagation feature, associated with spatial solitons in **power**- **law** media, constituting a new class of oscillatory solution, is identified.

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where V and M represent the variance and mean, respectively, of a series of specific ecological measures, for example, the population densities of insects in different-size quadrats, and α and β are constants to be estimated. This **power** **law** has been found to be valid in many areas of study [14]. Controversy surrounding recent relevant studies is about whether the exponent β of TPL is determined by an inherent biological mechanism or is only a purely statistical feature [15–18]. Cohen and Xu demonstrated that the estimate of the exponent of TPL is proportional to the skewness of the distribution generated by random sampling in blocks [15]. Using a feasible set approach, Xiao et al. showed that TPL arises from the constraining influences of two primary variables, namely, the number of individuals and the number of sites [16]. TPL with an exponent ranging from 1.0 to 2.0 could be generated by the most possible configurations of individuals among sites. Shi et al. [17] used two clustering point process models to examine the effect of dispersal distance of plant seeds or rhizomes on the estimate of the exponent of TPL, and found that with increasing dispersal distance the exponent described a complex response curve. In addition, dry weights of crops at different investigation times and the development and growth rates of insects at different temperatures have also been shown to follow TPL [18]. These measures associated with TPL appear to be related to the energy distribution among different statistical units. To use a more direct approach to demonstrate the link between TPL and energy distribution, Li et al. [19] checked the influence of different time interval divisions of the same seismic time series on the estimate of the exponent of TPL between the variance and mean of the released energies among different earthquakes within the same interval. The estimates of the exponent of TPL for the seismic time series approximated 2.0. Then, fixing the exponent of TPL to 2.0, the estimate of the logarithm of the constant α in TPL was demonstrated to be a logarithmic function of the time interval division. These studies related to TPL have added to our understanding of the variation of a large number of ecological and non-ecological measures, for example, animal and plant population density, biomass, poikilothermic developmental rate, crime, precipitation, released energies of aftershocks, trading activity of stock, and so on [13,18–22]. In our recent studies, weight is a typical representative of energy for biological measures, so the variance–mean relationship based on weight measures should reflect TPL well [23].

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Abstract—A simple adjustment to parametric failure-time distributions, which allows for much greater flexibility in the shape of the hazard-rate function, is considered. Analytical expressions for the distributions of the **power**-**law** adjusted Weibull, gamma, log-gamma, generalized gamma, lognormal and Pareto distributions are given. Most of these allow for bathtub shaped and other multi-modal forms of the hazard rate. The new distributions are fitted to real failure-time data which exhibit a multi-modal hazard-rate function and the fits are compared.

The **power** **law** behavior is found to various systems, starting from social sys- tems to brain protein-protein networks which indicate important functional and organizational characteristics. This behavior indicates that the properties of the system are independent of scale of the system. Since this **law** also reflects the fractal nature of the system, it characterizes as an indicator of self-organized be- havior of the material system. The impinging energetic electrons experience the self-organized behavior of the system which is reflected in the parameters calcu- lated using Monte Carlo simulation procedure. This idea of fractal nature could be used as an order parameter in the fabrication of multi-layered device with proper efficiency.

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inflation, the potential is approximately monomial. The family is parametrized by two real parameters, the optimal values of which are determined by contrasting the model with observations. We find that the predicted spectral index of the curvature perturbation is near the sweet spot of the Planck observations. For a sub-Planckian inflaton, the Lyth bound does not allow for a large value of the tensor to scalar ratio r. However, for mildly super-Planckian inflaton values, we obtain sizeable r, which is easily testable in the near future. Even though our treatment is phenomenological, and the form of the potential of **power**-**law** plateau inflation is data driven, we develop a simple toy model in global and local supersymmetry to demonstrate how **power**-**law** plateau infla- tion may well be realized in the context of fundamental theory. We use natural units, where c ¼ ℏ ¼ 1 and Newton ’ s gravitational constant is 8π G ¼ m −2 P , with m P ¼ 2.43 × 10 18 GeV being the reduced Planck mass.

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Taking into account the fact that the computer systems, as the implementations of Turing machine, are physical devices, the paper shows considerations in which hard drive behavior will be presented in terms of statistical mechanics. Be- cause computer is a machine, its analysis cannot be based only on mathematical models apart of physical conditions. In the paper it will be presented a very narrow part this problem—an analysis of hard drive behavior in the context of the **power**-**law** distributions. We will focus only on four selected hard drive parameters, i.e. the rate of transfer bytes to or from the disk during the read or write, the number of pending requests to the disk and the rate of read operations. Our research was performed under the Windows operating system and this allows to make a statistical analysis for the pos- sible occurrence of **power**-laws representing the lack of characteristic scale for considered processes. This property will be confirmed in all analyzed cases. A presented study can help describing the behavior of the whole computer sys- tem in terms of physics of computer processing.

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The **power** **law** relations that we have examined all display the tendency for the number of small events – Web sizes, links, populations, and GDP of small countries – to be less than what the rank- size rule predicts but with Simon’s (1957) model, this can easily be explained by the smaller domains having not yet reached maturity. We did not go as far as to compute growth rates or exponents for every level of rank but we did illustrate the plausibility of the hypothesis that the largest domains approximate the rank-size rule while the smaller domains are growing towards this steady state. The differences in **power** **law** that we computed between these two sets confirms this notion. In future work, we will explore these ideas further but to do this, we will require much better data at more than one point in time. This analysis based on a single time-point essentially forms a first step in an interpretation of how Web space is developing. There are many other issues and possibilities that need to be addressed herewith. As well as implementing a time-series analysis, we need to clarify definitions of domains in spatial as well as sectoral terms, and we need to consider suitable spatial and temporal aggregations which affect our analysis.

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In 1993, the seminal paper [3] (expanded in [4]) found evidence of the existence of **power** **law** relationships in network traffic by observing long- range correlation in Local Area Network (LAN) traffic. This brought the concept of self-similarity, and the related concept of Long-Range Dependence (LRD), into the field of network traffic and performance analysis. Before this finding, network traffic and performance studies had been mainly based on models, such as Poisson processes, which assume that traffic exhibits no long-term correlation. In networks with long-range correlated traffic, queuing performance can very different to that of traffic assumed independent or only having short-term correlations. Subsequently **power** **law** relationships have been observed in several other contexts on many different types of network. In 1999 it was also discovered that the global Internet structure is char- acterised by a **power** **law** [5]. That is, the probability distribution of a node’s connectivity (measured for example by the number of BGP peering relations that an autonomous system has) follows a **power** **law**. This discovery in- validated previous Internet models that were based on the classical random graphs. Since then a lot of efforts have been put into studying the Internet **power** **law** structure [6, 7, 8, 9, 10, 11, 12, 13, 14, 15].

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Inflation [5]) have received enormous attention. In these models (and similar other such models, e.g. α-attractors [6]), the approach to the inflationary plateau is exponential. In this case, however, distinguishing between models is difficult [7]. In contrast, a **power**-**law** inflationary plateau was considered in Ref. [8], where shaft inflation was introduced, based in global supersymmetry with a deceptively simple but highly non-perturbative superpotential W ∝ (Φ n + m n ) 1/n , where m is

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Analytical I-V models are necessary for the design of integrated circuits. The analytical treatment of MOSFET circuit is primarily done by Shockley model but this model, is not so much accurate, because it shows negligible effect for velocity saturation carriers and short channel effect. There have been many attempts to accurately model the characteristics of these transistors, including complicated empirical models used for SPICE simulations. After that analytic treatment of MOSFET is done by various precise MOS model like, spice level3 model, BSIM, etc. But, some of these takes more time in evaluating model, some needs a special system with a hardware/software combination to extracting model parameters. Some of these need expensive numerical iteration procedure to extract model parameters and extracted model parameters are not able to give satisfactory results. However, to fill the gap between the Shockley and more precise model a new model, named as nth-**power** model, preserving high accuracy is introduced for circuit analysis. The nth-**power** model (Sakurai and Newton, 1990; Sakurai and Newton, 1991), which assumes a non-integer nth-**power** relation between current and voltage, is the best model to extract parameters. The nth-**power** **law** MOSFET model is an extension of alpha **power** **law** MOSFET model but much more accurate in linear and saturation region.

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The measured values are plotted in Fig. 3. The measured circle on Phobos was the primary meridian (0 ◦ and 180 ◦ in longitude), and the measured line on Gaspra was from (35 ◦ N, 10 ◦ W) to (80 ◦ N, 190 ◦ W). The derived **power**-**law** indices were converted to values for the index φ of the fracture planes, using the results derived above. Both indices are nearly equal to 6, the fully cracked condition. Phobos’s smaller φ indicates that Phobos has been more fractured than Gaspra. The two values are as follows:

We develop semi-analytical, self-similar solutions for the oscillatory boundary layer (‘Stokes layer’) in a semi-infinite **power**-**law** fluid bounded by an oscil- lating wall (the so-called Stokes problem). These solutions differ significantly from the classical solution for a Newtonian fluid, both in the non-sinusoidal form of the velocity oscillations and in the manner at which their ampli- tude decays with distance from the wall. In particular, for shear-thickening fluids the velocity reaches zero at a finite distance from the wall, and for shear-thinning fluids it decays algebraically with distance, in contrast to the exponential decay for a Newtonian fluid. We demonstrate numerically that these semi-analytical, self-similar solutions provide a good approximation to the flow driven by a sinusoidally oscillating wall.

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In Figure 1a-e, we show the distribution of tenure lengths for managers in English football (soccer) clubs plotted logarithmically such that **power** laws are straight lines. The diagrams show the evolution over time: Figure 1a covers the period before 1900, Figures1b-1d cover successive periods and Figure 1e covers the period from 1874 until the present. We clearly see that the distribution evolves, and a **power** **law** emerges. We find a similar emerging pattern by just dividing the sample into four equal sized periods (not shown). This rules out the possibility that the pattern emerges purely because we add more data points. However, the number of managers fired within the first year of employment falls below the **power** **law**: too few managers are fired. Firing decisions within the first year (season) are likely to be determined by very different processes than decisions later in a manager’s career. When first hired, managers generally have a “honeymoon” period to build a team, so they are not really judged until their second season. Moreover, a typical contract includes clauses that make it expensive to fire a manager within a season; most market activity occurs between seasons. Our data is coded by allocating managers fired between the first and second year to the one year bin, managers fired between the second and third year to the two year bin, and so on. Since managers are rarely fired on the day they are

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A prime gap is the difference between two successive prime numbers. Prime gaps are casually thought to occur randomly. However, the “ k -tuple conjecture” suggests that prime gaps are non-random by estimating how often pairs, triples and larger group- ings of primes will appear. The k -tuple conjecture is yet to be proven, but a very re- cent work presents a result that contributes to a confirmation of the k -tuple conjec- ture by finding unexpected biases in the distribution of consecutive primes. Here, we present another contribution to confirmation of the k -tuple conjecture based on sta- tistical physics. The pattern we find comes in the form of a **power** **law** in the distribu- tion of prime gaps. We find that prime gaps are proportional to the inverse of the chance of a number to be prime.

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Abstract. Random graph theory is used to examine the “small-world phenomenon”– any two strangers are connected through a short chain of mutual acquaintances. We will show that for certain families of random graphs with given expected degrees, the average distance is almost surely of order log n/ log ˜ d where ˜ d is the weighted average of the sum of squares of the expected degrees. Of particular interest are **power** **law** random graphs in which the number of vertices of degree k is proportional to 1/k β for some fixed exponent β. For the case of β > 3, we prove that the average distance of the **power** **law** graphs is almost surely of order log n/ log ˜ d. However, many Internet, social, and citation networks are **power** **law** graphs with exponents in the range 2 < β < 3 for which the **power** **law** random graphs have average distance almost surely of order log log n, but have diameter of order log n (provided having some mild constraints for the average distance and maximum degree). In particular, these graphs contain a dense subgraph, that we call the core, having n c/ log log n vertices. Almost all vertices are within distance log log n of the core although there are vertices at distance log n from the core.

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Harking back to Equation 1 introduced in the previous section, for every point on the graph, the ‘y’, in the context of the aspects of the system of Test cricket, stands for the population of the class, while ‘x’ , in general, stands for the midpoint value of the class. In the best-fit line equation, for instance, for the number of Tests played, ‘y’ Test cricketers play ‘x’ Test matches each. Empty classes – those for which ‘y’ equals zero – are excluded from the regression. The data in each case (each class-width selection for each field) are regressed for three different trends – **power** **law**, quadratic polynomial and logarithmic – and these three are compared with each other on the basis of the R 2 value, which is the chief determinant of the

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There has been some previous work in characterizing the hi- erarchy of the Internet graph. In [13], the authors observed that ASes can be divided into four classes, with signiﬁcant variation of average degree between the classes. The authors propose a tiering induced upon the AS graph, with the class of ASes with the highest average degree at the top tier, followed by others in descending order of the average degree. A similar tiering of ASes has also been introduced in [12], [21], but using the logi- cal relationships between the ASes. In [12], the authors create a logical tiering based on the Customer-Provider relationship between ASes, with a Provider AS assigned a tier higher than its Customer. A similar approach has been suggested by [21], using logical relationships and some notion of node intercon- nectivity to distinguish adjacent tiers. A recent work [22] also considers the hierarchical characteristic of the Internet graph, and the degree to which the degree based generators capture this property. Their metric of choice is the distribution of link- values, where the value of a link can be roughly deﬁned as the number of node pairs whose shortest paths traverse this link. The authors study the distribution of link-values for the Inter- net router and AS level graphs, **power**-**law** graphs and some canonical graphs. Examining the distribution of this metric for different topologies and comparing it with the distribution of the classical tree topology, the authors qualitatively compare the relative degrees of hierarchy of graphs of different topolo- gies. Although this metric serves a useful purpose in distin- guishing among the different graphs, it falls short of answering several questions. The distribution of link-values does not lend any insight into the interconnection structure of the graph which would have led to this distribution. Also, observing the distri- bution of link-values does not answer some questions about the nature of the hierarchy, whether it is balanced, how deep is it etc. In other words, we are not able to “visualize” the structural properties of these graphs.

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Since the CMB probes a relatively small range of scales the errors in the running of the running from Planck are large, and values larger than typically expected from slow roll inflation are allowed. We therefore then used the flow formalism [47] to generate a large ensemble of infla- tion models, including some where the **power** spectrum is consistent with the CMB and LSS measurements on large scales and grows sufficiently with increasing k to form PBHs on small scales. In these cases we compared the 1st and 2nd order **power**-**law** parameterisations of the **power** spectrum with the calculation using the Stewart- Lyth expression [51]. Again we found that the **power**-**law** expansion led to amplitudes which were typically incor- rect by many orders of magnitude, and could be larger or smaller than the SL value. Since the abundance of PBHs formed depends exponentially on the amplitude of the

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; Where C and a are some constant; and a is scaling exponent. Normally the graphical representation of this equation results in skewed distribution (depending upon constant a). So, log-log graph is generally plotted, which results in linear line (with slope of graph near to 1). Various quantitative results are thus derived from these graphs and can be used to know the real world topologies more accurately. Presence of **power** **law** indicates that arbitrarily large values can occur with a non- negligible probability, and therefore, rather than ignoring these values as “outliers”, its statistical significance needs to be

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