Exponent Of Distribution Research Articles

Epistemic Modality: A Corpus-Based Analysis of Epistemic Markers in EU and Polish Judgments

AbstractThe aim of this paper is to establish the repertoire and distribution of verbal and adverbial exponents of epistemic modality in English- and Polish-language judgments passed by the Court of Justice of the EU (CJEU) and non-translated judgments passed by the Supreme Court of Poland (SN). The study applies a model for categorizing exponents of epistemicity with regard to their (i) level (high-, medium- and low-level of certainty, necessity or possibility expressed by the markers; primary dimension), (ii) perspective (own vs. reported perspective), (iii) opinion (based either on facts or beliefs) and (iv) time (the embedding of epistemic markers in sentences relating to the past, present or future) (contextual dimensions). It examines the degree of intra-generic convergence of translated EU judgments and non-translated national judgments in terms of the employment of epistemic markers, as well as the degree of authoritativeness of judicial argumentation, and determines whether the frequent use of epistemic markers constitutes a generic feature of judgments. The research material consists of a parallel corpus of English- and Polish-language versions of 200 EU judgments and a corpus of 200 non-translated domestic judgments. The results point to the high salience and differing patterns of use of epistemic markers in both EU and national judgments. The frequent use of high-level epistemic markers boosts the authoritativeness of judicial reasoning.

Intelligent Classification and Analysis of Essential Genes Using Quantitative Methods

Essential genes are considered to be the genes required to sustain life of different organisms. These genes encode proteins that maintain central metabolism, DNA replications, translation of genes, and basic cellular structure, and mediate the transport process within and out of the cell. The identification of essential genes is one of the essential problems in computational genomics. In this present study, to discriminate essential genes from other genes from a non-biologists perspective, the purine and pyrimidine distribution over the essential genes of four exemplary species, namely Homo sapiens , Arabidopsis thaliana , Drosophila melanogaster , and Danio rerio are thoroughly experimented using some quantitative methods. Moreover, the Indigent classification method has also been deployed for classification on the essential genes of the said species. Based on Shannon entropy, fractal dimension, Hurst exponent, and purine and pyrimidine bases distribution, 10 different clusters have been generated for the essential genes of the four species. Some proximity results are also reported herewith for the clusters of the essential genes.

Strong anomalous diffusion in two-state process with Lévy walk and Brownian motion

Strong anomalous diffusion phenomena are often observed in complex physical and biological systems, which are characterized by the nonlinear spectrum of exponents $q\nu(q)$ by measuring the absolute $q$-th moment $\langle |x|^q\rangle$. This paper investigates the strong anomalous diffusion behavior of a two-state process with L\'{e}vy walk and Brownian motion, which usually serves as an intermittent search process. The sojourn times in L\'{e}vy walk and Brownian phases are taken as power law distributions with exponents $\alpha_+$ and $\alpha_-$, respectively. Detailed scaling analyses are performed for the coexistence of three kinds of scalings in this system. Different from the pure L\'{e}vy walk, the phenomenon of strong anomalous diffusion can be observed for this two-state process even when the distribution exponent of L\'{e}vy walk phase satisfies $\alpha_+<1$, provided that $\alpha_-<\alpha_+$. When $\alpha_+<2$, the probability density function (PDF) in the central part becomes a combination of stretched L\'{e}vy distribution and Gaussian distribution due to the long sojourn time in Brownian phase, while the PDF in the tail part (in the ballistic scaling) is still dominated by the infinite density of L\'{e}vy walk.

A phase transition for preferential attachment models with additive fitness

Preferential attachment models form a popular class of growing networks, where incoming vertices are preferably connected to vertices with high degree. We consider a variant of this process, where vertices are equipped with a random initial fitness representing initial inhomogeneities among vertices and the fitness influences the attractiveness of a vertex in an additive way. We consider a heavy-tailed fitness distribution and show that the model exhibits a phase transition depending on the tail exponent of the fitness distribution. In the weak disorder regime, one of the old vertices has maximal degree irrespective of fitness, while for strong disorder the vertex with maximal degree has to satisfy the right balance between fitness and age. Our methods use martingale methods to show concentration of degree evolutions as well as extreme value theory to control the fitness landscape.

Fractal catastrophes

We analyse the spatial inhomogeneities (‘spatial clustering’) in the distribution of particles accelerated by a force that changes randomly in space and time. To quantify spatial clustering, the phase-space dynamics of the particles must be projected to configuration space. Folds of a smooth phase-space manifold give rise to catastrophes (‘caustics’) in this projection. When the inertial particle dynamics is damped by friction, however, the phase-space manifold converges towards a fractal attractor. It is believed that caustics increase spatial clustering also in this case, but a quantitative theory is missing. We solve this problem by determining how projection affects the distribution of finite-time Lyapunov exponents (FTLEs). Applying our method in one spatial dimension we find that caustics arising from the projection of a dynamical fractal attractor (‘fractal catastrophes’) make a distinct and universal contribution to the distribution of spatial FTLEs. Our results explain a projection formula for the spatial fractal correlation dimension, and how a fluctuation relation for the distribution of FTLEs for white-in-time Gaussian force fields breaks upon projection. We explore the implications of our results for heavy particles in turbulence, and for wave propagation in random media.

Climate persistence and memory

The autocorrelation function (ACF) and its relationship to fluctuation analysis (FA) are discussed, based on the reanalysis monthly mean geopotential height at 500 hPa from ECMWF (ERA-20C). ACF provides a measure of the influence of correlation at different time points. The principal pattern ACF shows relatively large positive correlations between every consecutive years in the central Asian continent and the North African continent. However, the confidence intervals of the ACF distributions are generally too wide, and so of low confidence level. It is found that the scaling rule of the ACF breaks down typically, while the distributions of the critical exponent of FA always show patterns. It is also found that the larger values of critical exponent appear in the tropical region, which reflects a larger climate persistence. The critical exponent becomes smaller in the jet stream regions, indicating less year to year reappearance. The characteristic time accounts for the weighted summation of all possible ACFs, which represents a memory length of a certain climate events. In the boreal winter, some robust patterns can be identified by the corresponding distribution of the characteristic time, which provides a unified way of describing the persistence of climate. An approximate relation between the critical exponent of FA and the characteristic time has been found, and such relation is verified by the distributions of these two physical variables.

Testing for non-chaoticity under noisy dynamics using the largest Lyapunov exponent

In this paper, we introduce a robust procedure to test for non-chaoticity when data are contaminated by an additive noise. Under the Kalman filter framework, our procedure first amounts to compute the largest Lyapunov exponent of the extracted signal. The exponent describes the log-divergence of a dynamical system (Rosenstein et al. in Phys D 65(1–2):117–134, 1993. https://doi.org/10.1016/0167-2789(93)90009-p). Then, using the so-called simulation smoother, we generate a high number of trajectories of the state-vector, conditional on the observed series, and compute the empirical distribution of the largest Lyapunov exponent. The distribution allows for computing confidence intervals. We can thus test if the largest Lyapunov exponent is not significantly greater than zero. Using Monte Carlo simulations, we show the validity of such an approach. We provide an illustration using toy models, which depict several dynamical systems. Finally, we implement tests of non-chaoticity on financial time series. We find no empirical evidence of chaotic patterns. Our approach is simple, efficient, and tests for chaos when data are measured with errors (i.e., noisy dynamics).

Universality of power-law exponents by means of maximum-likelihood estimation.

Power-law-type distributions are extensively found when studying the behavior of many complex systems. However, due to limitations in data acquisition, empirical datasets often only cover a narrow range of observation, making it difficult to establish power-law behavior unambiguously. In this work we present a statistical procedure to merge different datasets, with two different aims. First, we obtain a broader fitting range for the statistics of different experiments or observations of the same system. Second, we establish whether two or more different systems may belong to the same universality class. By means of maximum likelihood estimation, this methodology provides rigorous statistical information to discern whether power-law exponents characterizing different datasets can be considered equal among them or not. This procedure is applied to the Gutenberg-Richter law for earthquakes and for synthetic earthquakes (acoustic emission events) generated in the laboratory: labquakes. Different earthquake catalogs have been merged finding a Gutenberg-Richter law holding for more than eight orders of magnitude in seismic moment. The value of the exponent of the energy distribution of labquakes depends on the material used in the compression experiments. By means of the procedure proposed in this manuscript, we find that the Gutenberg-Richter law for earthquakes and charcoal labquakes can be characterized by the same power-law exponent, whereas Vycor labquakes exhibit a significantly different exponent.

Dye-tracer technique for rill flows by velocity profile measurements

Water flow on hillslope soil surface supplies energy which is required to detach soil particles, to transport and deposit sediments, therefore flow velocity is a key variable related to hillslope hydrodinamics of soil erosion processes. Among the different methods available for measuring velocity of shallow interrill and rill flow, the trace technique is widely used. Trace technique is applied by adding a material (salt, magnetic material, water isotope, floating object) and then measuring the speed of the material to travel a known distance from the injection point.When flow velocity is measured using a dye-tracing method, the mean velocity is calculated by multiplying the measured surface velocity of the leading edge of the tracer plume by a correction factor which was generally empirically deduced. The main uncertainty of the dye-tracing technique stands in the estimate of this correction factor, which is the ratio between the mean flow velocity and the surface velocity. The main goal of this paper is establishing a theoretical relationship for calculating the correction factor by using a power velocity profile. At first, the developed analysis demonstrated that the correction factor only depends on the exponent of the power velocity distribution (Eq. (5)). Then, this theoretical expression of the correction factor was applied using the velocity profiles measured by Baiamonte and Ferro for the condition of sediment-free flow in motion on a rough bed and by Coleman for a sediment-laden flow moving on a smooth flume.Using the correction factor values calculated by the velocity measurements carried out for the sediment-free flow in motion on a rough bed, the relationship between the correction factor and roughness height was established. In agreement with a previous study by Ali et al., for a sediment-free flow the correction factor increases with roughness height.Finally, the velocity profile measurements carried out for a sediment-laden flow in motion on a smooth bed were used to state the effect of sediment load on the correction factor. This last analysis allowed to conclude, in agreement with Li and Abrahams and Zhang et al., that the correction factor decreases when the sediment load increases.

STRANGE NONCHAOTIC ATTRACTORS IN A QUASIPERIODICALLY-FORCED PIECEWISE SMOOTH SYSTEM WITH FAREY TREE

The existence of strange nonchaotic attractors (SNAs) is verified in a simple quasiperiodically-forced piecewise smooth system with Farey tree. It can be seen that more and more jumping discontinuities appear on the smooth torus and the torus becomes extremely fragmented with the change of control parameter. Finally, the torus becomes an SNA with fractal property. In order to confirm the existence of SNAs in this system, we preliminarily use the estimation of the phase sensitivity exponent, estimation of the largest Lyapunov exponent and rational approximation. SNAs are further characterized by power spectra, recurrence plots, the largest Lyapunov exponents and their variance, the distribution of the finite-time Lyapunov exponents, the spectral distribution function and scaling laws.

The regular dynamics through the finite‐time Lyapunov exponent distributions in 3D Hamiltonian systems

AbstractWe investigate the application as a technique of the so‐called finite‐time Lyapunov exponent, a scalar value that measures the average, or integrated, separation between trajectories in integrable dynamical systems to quantify the predictability limit of the asymptotic global behavior of old compact star systems (1–10 Gyr), for instance, neutron stars. For this purpose, the Hamiltonian systems with a three‐dimensional self‐gravitating axisymmetric potential are superimposed due to the influence of different birth and kick velocity distributions. We find the behavior of the density distributions P(x) for the finite‐time exponents, which follow a log‐normal distribution (transformed into a Gaussian distribution), with a given mean deviation centered around the global values, at larger Δt. It is shown that simple models of distributions reflect the underlying dynamics. We also analyze the predictability and the deviation vector ω(t), obtained from the distributions, by randomly selecting the initial deviation directions, and consequently the stellar dynamic stability of the periodic orbits, in three‐dimensional (3D) Hamiltonian systems over time. The behavior of the orbits is strongly influenced by the regular nature of the motion.

Analysis of human behavior statistics law based on WeChat Moment

WeChat is one of the most popular social platforms where users share their feelings and opinions in Moments, Wechat postings that are usually in forms of text, pictures and videos, etc. This paper analyze the Moment posting behavior of WeChat users by obtaining 10196 Moments of 628 users. According to the result, both the number of commits and the inter-event time conform to the power-law distribution in a continuous period of time, revealing a fact that the majority of Moments are posted by a small portion of users. Then our study on individual activity unveils a relatively strong positive correlation between user activity and the exponent of inter-event time distribution, which agrees with previous studies. Additionally, the paper points out that WeChat users have a low-burstiness, low-memory posting pattern, which is slightly different from that of other online systems owing to Wechat users’ generally high activity. This paper also studied the effects of sleep on user behavior. Our findings have significance in understanding the human dynamic mechanism of Wechat users.

Information production in homogeneous isotropic turbulence.

We study the Reynolds number scaling of the Kolmogorov-Sinai entropy and attractor dimension for three-dimensional homogeneous isotropic turbulence through the use of direct numerical simulation. To do so, we obtain Lyapunov spectra for a range of different Reynolds numbers by following the divergence of a large number of orthogonal fluid trajectories. We find that the attractor dimension grows with the Reynolds number as Re^{2.35} with this exponent being larger than predicted by either dimensional arguments or intermittency models. The distribution of Lyapunov exponents is found to be finite around λ≈0 contrary to a possible divergence suggested by Ruelle. The relevance of the Kolmogorov-Sinai entropy and Lyapunov spectra in comparing complex physical systems is discussed.

SPATIAL DISTRIBUTION OF Chrysodeixis includens EGGS (WALKER, 1858) (LEPIDOPTERA: NOCTUIDAE) IN SOYBEAN CROP

Among pest insects that attack soybean crops, Chrysodeixis includens (Walker, 1858) (Lepidoptera: Noctuidae) looper caterpillar deserves attention due to its key pest status in soybean crops. The spatial distribution of C. includens eggs should be investigated in order to understand the behavior of this species in the area. The aim of this study was to investigate the spatial distribution of C. includens eggs in soybean crops. The experiment was conducted with SYN 9070 RR soybean variety in an experimental area of the Teaching, Research and Extension Farm (FEPE) of FCAV/ UNESP, Jaboticabal, SP, Brazil. The area of 0.6 ha was divided into 60 equidistant plots of 100 m² each. For the study of the spatial distribution of P. includens eggs in the area, the following dispersion indexes were used: variance / mean ratio (I), Morisita index (Id), Green coefficient (Cx), k exponent of the negative binomial distribution for each sampling. Regarding the probabilistic models that describe the spatial distribution of a given variable, the data adjustment to the Poisson and negative binomial distributions was tested. According to values obtained for dispersion indexes, egg distribution occurred in an aggregate way, and the negative binomial distribution was the most appropriate probabilistic model to represent the distribution of C. includens eggs in the study area.

Neuronal avalanches and time-frequency representations in stimulus-evoked activity

Neuronal avalanches are a hallmark feature of critical dynamics in the brain. While the theoretical framework of a critical branching processes is generally accepted for describing avalanches during ongoing brain activity, there is a current debate about the corresponding dynamical description during stimulus-evoked activity. As the brain activity evoked by external stimuli considerably varies in magnitude across time, it is not clear whether the parameters that govern the neuronal avalanche analysis (a threshold or a temporal scale) should be adaptively altered to accommodate these changes. Here, the relationship between neuronal avalanches and time-frequency representations of stimulus-evoked activity is explored. We show that neuronal avalanche metrics, calculated under a fixed threshold and temporal scale, reflect genuine changes in the underlying dynamics. In particular, event-related synchronization and de-synchronization are shown to align with variations in the power-law exponents of avalanche size distributions and the branching parameter (neural gain), as well as in the spatio-temporal spreading of avalanches. Nonetheless, the scale-invariant behavior associated with avalanches is shown to be a robust feature of healthy brain dynamics, preserved across various periods of stimulus-evoked activity and frequency bands. Taken together, the combined results suggest that throughout stimulus-evoked responses the operating point of the dynamics may drift within an extended-critical-like region.

Research on the Behavior of the Shill Bidding with Affiliated Values

On-line auction provides convenience, as well as causes the frequent occurrence of many deceits, the passage mainly provides the concealed conditions for the seller’s shill bidding behavior based on the anonymity of on-line auction, it discusses the bidder’s optimal decision quotation when the shill bidding behavior happens or nor in the affiliated value pattern. To establish the bidder and seller’s incomplete information dynamic game model when occurring the shill bidding and analyze the probability of shill bidding can provide the reference for bidder’s quotation. To consider the time, reserved price (price accepted by the seller) and the current highest auction price which affect the shill bidding and match the relevant nonhomogeneous poisson process and exponent distribution can provide the reference for bidders to judge if the shill bidding behavior occurs by sellers.

Probabilistic Caching for Small-Cell Networks With Terrestrial and Aerial Users

The support for aerial users has become the focus of recent 3GPP standardizations of 5G, due to their high maneuverability and flexibility for on-demand deployment. In this paper, probabilistic caching is studied for ultra-dense small-cell networks with terrestrial and aerial users, where a dynamic on-off architecture is adopted under a sophisticated path loss model incorporating both line-of-sight and non-line-of-sight transmissions. Generally, this paper focuses on the successful download probability (SDP) of user equipments (UEs) from small-cell base stations (SBSs) that cache the requested files under various caching strategies. To be more specific, the SDP is first analyzed using stochastic geometry theory, by considering the distribution of such two-tier UEs and SBSs as Homogeneous Poisson Point Processes. Second, an optimized caching strategy (OCS) is proposed to maximize the average SDP. Third, the performance limits of the average SDP are developed for the popular caching strategy (PCS) and the uniform caching strategy (UCS). Finally, the impacts of the key parameters, such as the SBS density, the cache size, the exponent of Zipf distribution and the height of aerial user, are investigated on the average SDP. The analytical results indicate that the UCS outperforms the PCS if the SBSs are sufficiently dense, while the PCS is better than the UCS if the exponent of Zipf distribution is large enough. Furthermore, the proposed OCS is superior to both the UCS and PCS.

Power Law of Crack Length Distribution in the Multiple Damage Process

Multiple fatigue damage, which is characterized by crack initiation and propagation processes, is considered. We proposed two models of multiple damage, which imply random crack initiation and further propagation, with the exponential dependence between their length on the number of loading cycles. Crack initiation is modeled by the stationary Poisson flow with a constant intensity, while crack propagation is characterized by the rate parameter controlling the dependence of crack propagation rate and its length. The first model describes the deterministic case of multiple crack propagation at a fixed value of the above rate parameter, while the second one predicts their propagation by random trajectories according to distribution of the rate parameter. In the former case, crack length distribution is shown to be the Pareto power law with the exponent, which is defined by the ratio of kinetic parameters of initiation and propagation of defects. In the latter case, the rate parameter is uniformly distributed, in accordance with experimental data, so that the power-law distribution of crack length is close to the Pareto distribution. The respective distribution exponent also depends on the ratio of kinetic parameters of multiple damages and tends to drop during damage accumulation to the threshold level (namely, reaches the value of 2). This finding complies with experimental data on multiple damages of various classes of materials, including metals and rock masses. We also substantiated the range of ratios of kinetic parameters of defect initiation and propagation, which ensure the Pareto law of cracks length distribution and can be used to estimate the critical state of cracked bodies.

Inference on Long-Range Temporal Correlations in Human EEG Data

Detrended Fluctuation Analysis (DFA) is a statistical estimation algorithm used to assess long-range temporal dependence in neural time series. The algorithm produces a single number, the DFA exponent, that reflects the strength of long-range temporal correlations in the data. No methods have been developed to generate confidence intervals for the DFA exponent for a single time series segment. Thus, we present a statistical measure of uncertainty for the DFA exponent in electroencephalographic (EEG) data via application of a moving-block bootstrap (MBB). We tested the effect of three data characteristics on the DFA exponent: (1) time series length, (2) the presence of artifacts, and (3) the presence of discontinuities. We found that signal lengths of ∼5 minutes produced stable measurements of the DFA exponent and that the presence of artifacts positively biased DFA exponent distributions. In comparison, the impact of discontinuities was small, even those associated with artifact removal. We show that it is possible to combine a moving block bootstrap with DFA to obtain an accurate estimate of the DFA exponent as well as its associated confidence intervals in both simulated data and human EEG data. We applied the proposed method to human EEG data to (1) calculate a time-varying estimate of long-range temporal dependence during a sleep-wake cycle of a healthy infant and (2) compare pre- and post-treatment EEG data within individual subjects with pediatric epilepsy. Our proposed method enables dynamic tracking of the DFA exponent across the entire recording period and permits within-subject comparisons, expanding the utility of the DFA algorithm by providing a measure of certainty and formal tests of statistical significance for the estimation of long-range temporal dependence in neural data.

The Order of the Dirichlet Series in the Half-Strip

We study the Dirichlet series whose domain of convergence is the half-plane, and the sequence of exponents can be extended to a certain “regular” sequence. We prove that the k-orders of the Dirichlet series are the same in all the half-strips whose widths are greater than a certain number called the special density of the distribution of exponents.