Menzerath–Altmann law for distinct word distribution analysis in a large text

The empirical law uncovered by Menzerath and formulated by Altmann, known as the Menzerath–Altmann law (henceforth the MA law), reveals the statistical distribution behavior of human language in various organizational levels. Building on previous studies relating organizational regularities in a language, we propose that the distribution of distinct (or different) words in a large text can effectively be described by the MA law. The validity of the proposition is demonstrated by examining two text corpora written in different languages not belonging to the same language family (English and Turkish). The results show not only that distinct word distribution behavior can accurately be predicted by the MA law, but that this result appears to be language-independent. This result is important not only for quantitative linguistic studies, but also may have significance for other naturally occurring organizations that display analogous organizational behavior. We also deliberately demonstrate that the MA law is a special case of the probability function of the generalized gamma distribution.     

Excess entropy in natural language: Present state and perspectives

We review recent progress in understanding the meaning of mutual information in natural language. Let us define words in a text as strings that occur sufficiently often. In a few previous papers, we have shown that a power-law distribution for so defined words (a.k.a. Herdan's law) is obeyed if there is a similar power-law growth of (algorithmic) mutual information between adjacent portions of texts of increasing length. Moreover, the power-law growth of information holds if texts describe a complicated infinite (algorithmically) random object in a highly repetitive way, according to an analogous power-law distribution. The described object may be immutable (like a mathematical or physical constant) or may evolve slowly in time (like cultural heritage). Here, we reflect on the respective mathematical results in a less technical way. We also discuss feasibility of deciding to what extent these results apply to the actual human communication.     

Modeling long-range cross-correlations in two-component ARFIMA and FIARCH processes

We investigate how simultaneously recorded long-range power-law correlated multivariate signals cross-correlate. To this end we introduce a two-component ARFIMA stochastic process and a two-component FIARCH process to generate coupled fractal signals with long-range power-law correlations which are at the same time long-range cross-correlated. We study how the degree of cross-correlations between these signals depends on the scaling exponents characterizing the fractal correlations in each signal and on the coupling between the signals. Our findings have relevance when studying parallel outputs of multiple component of physical, physiological and social systems.     

Physical model of dimensional regularization

We explicitly construct fractals of dimension \(4{-}\varepsilon \) on which dimensional regularization approximates scalar-field-only quantum-field theory amplitudes. The construction does not require fractals to be Lorentz-invariant in any sense, and we argue that there probably is no Lorentz-invariant fractal of dimension greater than 2. We derive dimensional regularization’s power-law screening first for fractals obtained by removing voids from 3-dimensional Euclidean space. The derivation applies techniques from elementary dielectric theory. Surprisingly, fractal geometry by itself does not guarantee the appropriate power-law behavior; boundary conditions at fractal voids also play an important role. We then extend the derivation to 4-dimensional Minkowski space. We comment on generalization to non-scalar fields, and speculate about implications for quantum gravity.     

Classification of Literary Works: Fractality and Complexity of the Narrative,Essay, and Research Article

A complex network as an abstraction of a language system has attracted much attention during the last decade. Linguistic typological research using quantitative measures is a current research topic based on the complex network approach. This research aims at showing the node degree, betweenness, shortest path length, clustering coefficient, and nearest neighbourhoods’ degree, as well as more complex measures such as: the fractal dimension, the complexity of a given network, the Area Under Box-covering, and the Area Under the Robustness Curve. The literary works of Mexican writers were classify according to their genre. Precisely 87% of the full word co-occurrence networks were classified as a fractal. Also, empirical evidence is presented that supports the conjecture that lemmatisation of the original text is a renormalisation process of the networks that preserve their fractal property and reveal stylistic attributes by genre.     

From 1/f noise to multifractal cascades in heartbeat dynamics

We explore the degree to which concepts developed in statistical physics can be usefully applied to physiological signals. We illustrate the problems related to physiologic signal analysis with representative examples of human heartbeat dynamics under healthy and pathologic conditions. We first review recent progress based on two analysis methods, power spectrum and detrended fluctuation analysis, used to quantify long-range power-law correlations in noisy heartbeat fluctuations. The finding of power-law correlations indicates presence of scale-invariant, fractal structures in the human heartbeat. These fractal structures are represented by self-affine cascades of beat-to-beat fluctuations revealed by wavelet decomposition at different time scales. We then describe very recent work that quantifies multifractal features in these cascades, and the discovery that the multifractal structure of healthy dynamics is lost with congestive heart failure. The analytic tools we discuss may be used on a wide range of physiologic signals. (c) 2001 American Institute of Physics.     

Anomalously large critical regions in power-law random matrix ensembles

We investigate numerically the power-law random matrix ensembles. Wave functions are fractal up to a characteristic length whose logarithm diverges asymmetrically with different exponents, 1 in the localized phase and 0.5 in the extended phase. The characteristic length is so anomalously large that for macroscopic samples there exists a finite critical region, in which this length is larger than the system size. The Green's functions decrease with distance as a power law with an exponent related to the correlation dimension.     

Word diversity can accelerate consensus in naming game

In this paper, we introduce word diversity that reflects the inhomogeneity of words in a communication into the naming game. Diversity is realized by assigning a weight factor to each word. The weight is determined by three different distributions (uniform, exponential, and power-law distributions). During the communication, the probability that a word is selected from speaker's memory depends on the introduced word diversity. Interestingly, we find that the word diversity following three different distributions can remarkably promote the final convergency, which is of high importance in the self-organized system. In particular, for all the ranges of amplitude of distribution, the power-law distribution enables the fastest consensus, while uniform distribution gives the slowest consensus. We provide an explanation of this effect based on both the number of different names and the number of total names, and find that a wide spread of names induced by the segregation of words is the main promotion factor. Other quantities, including the evolution of the averaging success rate of negotiation and the scaling behavior of consensus time, are also studied. These results are helpful for better understanding the dynamics of the naming game with word diversity.     

Word diversity can accelerate consensus in naming game

In this paper,we introduce word diversity that reflects the inhomogeneity of words in a communication into the naming game.Diversity is realized by assigning a weight factor to each word.The weight is determined by three different distributions(uniform,exponential,and power-law distributions).During the communication,the probability that a word is selected from speaker’s memory depends on the introduced word diversity.Interestingly,we find that the word diversity following three different distributions can remarkably promote the final convergency,which is of high importance in the self-organized system.In particular,for all the ranges of amplitude of distribution,the powerlaw distribution enables the fastest consensus,while uniform distribution gives the slowest consensus.We provide an explanation of this effect based on both the number of different names and the number of total names,and find that a wide spread of names induced by the segregation of words is the main promotion factor.Other quantities,including the evolution of the averaging success rate of negotiation and the scaling behavior of consensus time,are also studied.These results are helpful for better understanding the dynamics of the naming game with word diversity.     

Fractal power law in literary English

We present in this paper a numerical investigation of literary texts by various well-known English writers, covering the first half of the twentieth century, based upon the results obtained through corpus analysis of the texts. A fractal power law is obtained for the lexical wealth defined as the ratio between the number of different words and the total number of words of a given text. By considering as a signature of each author the exponent and the amplitude of the power law, and the standard deviation of the lexical wealth, it is possible to discriminate works of different genres and writers and show that each writer has a very distinct signature, either considered among other literary writers or compared with writers of non-literary texts. It is also shown that, for a given author, the signature is able to discriminate between short stories and novels.     

Fractal Analysis of Power-Law Fluid in a Single Capillary

The fractai expressions for flow rate and hydraulic conductivity for power-law fluids in a single capillary are derived based on the fractai nature of tortuous capillaries. Every parameter in the proposed expressions has clear physical meaning. The flow rate and hydraulic conductivity for power-law fluids are found to be related to the tortuosity fractal dimension and the power-law index. The flow rate for power-law fluids increases with the increasing power-law index but decreases with the increasing tortuosity fractal dimension. Good agreement between the model predictions for flow in a fractai capillary and in a converging-diverging duct is obtained. The results suggest that the fractal capillary model can be used to model the power-law fluids with different rheologicai properties.     

A network of two-Chinese-character compound words in the Japanese language

Some statistical properties of a network of two-Chinese-character compound words in the Japanese language are reported. In this network, a node represents a Chinese character and an edge represents a two-Chinese-character compound word. It is found that this network has properties of being “small-world” and “scale-free”. A network formed by only Chinese characters for common use (joyo-kanji in Japanese), which is regarded as a subclass of the original network, also has the small-world property. However, a degree distribution of the network exhibits no clear power law. In order to reproduce the disappearance of the power-law property, a model for a selecting process of the Chinese characters for common use is proposed.     

Localization of elastic waves in heterogeneous media with off-diagonal disorder and long-range correlations

Using the Martin-Siggia-Rose method, we study propagation of acoustic waves in strongly heterogeneous media which are characterized by a broad distribution of the elastic constants. Gaussian-white distributed elastic constants, as well as those with long-range correlations with nondecaying power-law correlation functions, are considered. The study is motivated in part by a recent discovery that the elastic moduli of rock at large length scales may be characterized by long-range power-law correlation functions. Depending on the disorder, the renormalization group (RG) flows exhibit a transition to localized regime in any dimension. We have numerically checked the RG results using the transfer-matrix method and direct numerical simulations for one- and two-dimensional systems, respectively.     

Statistical keyword detection in literary corpora

Understanding the complexity of human language requires an appropriate analysis of the statistical distribution of words in texts. We consider the information retrieval problem of detecting and ranking the relevant words of a text by means of statistical information referring to the spatial use of the words. Shannon's entropy of information is used as a tool for automatic keyword extraction. By using The Origin of Species by Charles Darwin as a representative text sample, we show the performance of our detector and compare it with another proposals in the literature. The random shuffled text receives special attention as a tool for calibrating the ranking indices.     

Sound scattering by random fractal inhomogeneities in the ocean

Sound scattering by random volume inhomogeneities (fluctuations of the refraction index in a medium) with an arbitrary anisotropy is considered using the small perturbation method (Born’s approximation). Surfaces (boundaries) of the inhomogeneities are deemed to be fractal ones: the energy spectra of the refraction index fluctuations follow the power law with a nonintegral exponent. Formulas are obtained for the volume scattering coefficient. Frequency and angular dependences of the scattering coefficient and their relations to the fractal dimension of inhomogeneities with different kinds of anisotropy and different sizes (on the sound wavelength scale) are presented. The fractal dimension of the inhomogeneities is estimated.     

Small-angle scattering from fat fractals

A number of experimental small-angle scattering (SAS) data are characterized by a succession of power-law decays with arbitrarily decreasing values of scattering exponents. To describe such data, here we develop a new theoretical model based on 3D fat fractals (sets with fractal structure, but nonzero volume) and show how one can extract structural information about the underlying fractal structure. We calculate analytically the monodisperse and polydisperse SAS intensity (fractal form factor and structure factor) of a newly introduced model of fat fractals and study its properties in momentum space. The system is a 3D deterministic mass fractal built on an extension of the well-known Cantor fractal. The model allows us to explain a succession of power-law decays and respectively, of generalized power-law decays (GPLD; superposition of maxima and minima on a power-law decay) with arbitrarily decreasing scattering exponents in the range from zero to three. We show that within the model, the present analysis allows us to obtain the edges of all the fractal regions in the momentum space, the number of fractal iteration and the fractal dimensions and scaling factors at each structural level in the fractal. We applied our model to calculate an analytical expression for the radius of gyration of the fractal. The obtained quantities characterizing the fat fractal are correlated to variation of scaling factor with the iteration number.     

Order-fractal transitions in abstract paintings

In this study, we determined the degree of order for 22 Jackson Pollock paintings using the Hausdorff–Besicovitch fractal dimension. Based on the maximum value of each multi-fractal spectrum, the artworks were classified according to the year in which they were painted. It has been reported that Pollock’s paintings are fractal and that this feature was more evident in his later works. However, our results show that the fractal dimension of these paintings ranges among values close to two. We characterize this behavior as a fractal-order transition. Based on the study of disorder-order transition in physical systems, we interpreted the fractal-order transition via the dark paint strokes in Pollock’s paintings as structured lines that follow a power law measured by the fractal dimension. We determined self-similarity in specific paintings, thereby demonstrating an important dependence on the scale of observations. We also characterized the fractal spectrum for the painting entitled Teri’s Find. We obtained similar spectra for Teri’s Find and Number 5, thereby suggesting that the fractal dimension cannot be rejected completely as a quantitative parameter for authenticating these artworks.     

分形油藏非牛顿幂律流体低速非达西不稳定渗流的组合数学模型

结合分形理论与渗流理论,对分形油藏非牛顿幂律流体低速非达西不稳定渗流的试井分析问题的数学模型进行了推导.该分形油藏模型由内域为非牛顿幂律流体低速非达西渗流,外域为非牛顿幂律流体达西渗流的同心圆域组成.在考虑井筒储存、表皮效应影响下,建立了该油藏的不稳定渗流有效井径组合数学模型,在3种外边界条件下求出了两个区域内压力在Laplace空间的解析解,应用Stehfest数值反演方法求得井底的无因次压力,分析了井底压力动态特征和参数影响.非牛顿幂律流体的幂律指数、分形参数均对典型曲线产生较大的影响,呈现出与牛顿流体和均质油藏明显不同的特征.这对非均质油藏非牛顿流体的不稳定试井分析及研究其非线性渗流特征均十分重要.     

Power law and multiscaling properties of the Chinese stock market

We investigate the cumulative probability density function (PDF) and the multiscaling properties of the returns in the Chinese stock market. By using returns data adjusted for thin trading, we find that the distribution has power-law tails at shorter microscopic timescales or lags. However, the distribution follows an exponential law for longer timescales. Furthermore, we investigate the long-range correlation and multifractality of the returns in the Chinese stock market by the DFA and MFDFA methods. We find that all the scaling exponents are between 0.5 and 1 by DFA method, which exhibits the long-range power-law correlations in the Chinese stock market. Moreover, we find, by MFDFA method, that the generalized Hurst exponents h(q) are not constants, which shows the multifractality in the Chinese stock market. We also find that the correlation of Shenzhen stock market is stronger than that of Shanghai stock market.