Mandelbrot Law Research Articles

N-convexity and weighted majorization with applications to f-divergences and Zipf–Mandelbrot law

n-convexity and weighted majorization with applications to f-divergences and Zipf–Mandelbrot law

Fundamental limits to economic development in developing and underdeveloped countries imposed by global hierarchy

Despite great leaps in science, technology, and human civilization since World War II, many countries in the world have been stuck in “middle income” traps or remained hopelessly poor. To understand this reality,we introduce a new metric, the revealed comparative wealth (RCW), and develop a theory of global hierarchical order (GHO), which is characterized by a Zipf–Mandelbrot law (ZML) for the yearly ranked RCW for all the countries in the world. Major factors contributing to the existence of GHO have been identified. Various types of behavior, including equilibria characterized by largely constant RCW, strongly coupled advanced economies modulated by business cycles, strongly competing developing and underdeveloped economies opposite-phase locked to advanced economies, and steadily growing or decaying economies synergistically interact to create ZML. The mathematical structure of ZML imposes fundamental limits to world economic development and sheds light on why “middle income” traps have persisted.

Energy-Constrained Random Walk with Boundary Replenishment

We study an energy-constrained random walker on a length-N interval of the one-dimensional integer lattice, with boundary reflection. The walker consumes one unit of energy for every step taken in the interior, and energy is replenished up to a capacity of M on each boundary visit. We establish large N, M distributional asymptotics for the lifetime of the walker, i.e., the first time at which the walker runs out of energy while in the interior. Three phases are exhibited. When M ll N^2 (energy is scarce), we show that there is an M-scale limit distribution related to a Darling–Mandelbrot law, while when M gg N^2 (energy is plentiful) we show that there is an exponential limit distribution on a stretched-exponential scale. In the critical case where M / N^2 rightarrow rho in (0,infty ), we show that there is an M-scale limit in terms of an infinitely-divisible distribution expressed via certain theta functions.

Generalizations of Levinson type inequalities via new Green functions with applications to information theory

In this paper, four new Green functions are used to generalize Levinson-type inequalities for the class of 3-convex functions. The f-divergence, Renyi entropy, Renyi divergence, Shannon entropy, and the Zipf–Mandelbrot law are also used to apply the main results to information theory.

Wavelet Analysis of Average Monthly Temperature New Delhi 1931- 2021 and Forecast until 2110

The identification method in the CurveExpert-1.40 software environment revealed asymmetric wavelets of changes in the average monthly temperature of New Delhi from 1931 to 2021. The maximum increment for 80 years of the average monthly temperature of 5.1°C was in March 2010. An analysis of the wave patterns of the dynamics of the average monthly temperature up to 2110 was carried out. For forecasting, formulas were adopted containing four components, among which the second component is the critical heat wave of India. The first component is the Mandelbrot law (in physics). It shows the natural trend of decreasing temperature. The second component increases according to the critical law. The third component with a correlation coefficient of 0.9522 has an annual fluctuation cycle. The fourth component with a semi-annual cycle shows the influence of vegetation cover. The warming level of 2010 will repeat again in 2035-2040. From 2040 the temperature will rise steadily. June is the hottest month. At the same time, the maximum temperature of 35.1°C in 2010 in June will again reach by 2076. But according to the second component of the heat wave, the temperature will rise from 0.54°C to 16.29°C. The annual and semi-annual cycles had an insignificant effect on the June temperature dynamics. Thus, the identification method on the example of meteorological observations in New Delhi made it possible to obtain summary models containing a different number of components. The temperature at a height of 2 m is insufficient. On the surface, according to space measurements, the temperature reaches 55°C. As a result, in order to identify more accurate asymmetric wavelets for forecasting, the results of satellite measurements of the surface temperature of India at various geographical locations of meteorological stations are additionally required.

On the size distribution of truncation areas for fractal surfaces

The size distribution of contact spot areas is critical for the evaluation of contact behavior of rough surfaces. A size distribution law proposed by Mandelbrot has been used widely in multi-asperity fractal contact models to predict the rough contact behavior. However, the Mandelbrot law has a limitation on the representation of the truncation area distribution for rough surfaces. Therefore, the present study develops a new size distribution law of the truncation areas for fractal surfaces. Specifically, the three-dimensional Weierstrass-Mandelbrot (WM) function is used to characterize rough surfaces. A novel method is proposed to determine the fractal parameters of the WM function. Numerical investigation of rough surface truncation is conducted to obtain size distribution information of the truncation regions for various rough surfaces. Based on the numerical results, the new size distribution law of the truncation areas is proposed and validated by fractal surface and realistic rough surface. It is found that the proposed law can accurately reproduce the size distribution of the truncation areas for rough surfaces with a wide range of fractal parameters.

Asymmetric Mean Annual Temperature Wavelets Surface Air Layer of Berlin for 1701–2021

The regularities of the dynamics of the average annual temperature of Berlin from 1701 to 2021 are revealed. A total of 65 wavelets were received. The temperature has a high quantum certainty, and the change in the average annual temperature of Berlin was identified by a model that contains only two components for prediction. The basis of the forecast at 320 years makes it possible to look into the future until the year 2340. The forecast confirms the conclusions made in the CMIP5 report on global warming. With an increase in the number of components in the model up to five, the forecast is possible only until 2060. Therefore, the model with only two components is workable. The trend is characterized by a modified Mandelbrot equation showing exponential growth with a high growth rate of 1.47421. The wave equation also has an amplitude in the form of the Mandelbrot law (in mathematics, the Laplace law, in biology, the Zipf-Pearl law, in econometrics, the Pareto law), when the exponential growth activity is equal to 1. For 1701, the period of oscillation was 2× 60.33333 ≈ 120.7 years. By 2021, the period decreased and became equal to 87.6 years. The trend is such that by 2340 the period of oscillation will decrease to 30.2 years. Such an increase in fluctuations indicates an imbalance in climate disturbances in temperature in Berlin. For Berlin, the last three years are characterized by sharp decreases in the average annual temperature from 11.8 °C to 10.5 °C, i.e. by 12.4% in 2021. Therefore, the forecast is still unstable, as a further decrease in the average annual temperature of Berlin in the near future may change the picture of the forecast.

On the Converse Jensen-Type Inequality for Generalized f-Divergences and Zipf–Mandelbrot Law

Motivated by some recent investigations about the sharpness of the Jensen inequality, this paper deals with the sharpness of the converse of the Jensen inequality. These results are then used for deriving new inequalities for different types of generalized f-divergences. As divergences measure the differences between probability distributions, these new inequalities are then applied on the Zipf–Mandelbrot law as a special kind of a probability distribution.

Environmental Distribution of Land Plots at Subjects of Russia

It is proposed to identify the hierarchy of the subjects of the federation according to the ecological possibilities of the territory according to places in the ranking according to the sum of the ranks of the fraction of the area of 12 types of land from the land area (total area without land under water). For a scientifically substantiated change in the composition of the territories, it is necessary to iden-tify the patterns of distributions of all types of land in the subjects of the Russian Federation. For this, it should be recognized that the population is not only a person, but also plants (wild and cultivated) and animals (wild and domestic). The article presents trends based on the modified Laplace law (in mathematics), Mandelbrot (in physics), Zipf-Pearl (in biology) and Pareto (in econometrics). The graphs of multiple fractal distribution of the land share according to the modified Mandelbrot law are shown. Due to human intervention, oscillations asymmetric in amplitude and period (wavelet signals) appear in addition to the trend. Together, trend and wavelets give an adequacy in correlation coefficient of more than 0.99.

Statistical tests for text homogeneity: using forward and backward processes of numbers of different words

The processes of growth in the number of diverse words in a text, when reading in the forward and backward directions, are studied in this article. Based upon the statistics achieved from the difference between these two processes, we construct a statistical test. This statistical test is used for text homogeneity checks. The elementary model states that words in a text are selected from some dictionary independent of each other according to the Zipf–Mandelbrot law. P-values of the statistical test are calculated based on the elementary probabilistic model using the asymptotic normality of corresponding statistics. At last but not least, this statistical test is applied for the analysis of homogeneity of sequences of sonnets.

Zipf’s law for cosmic structures: How large are the greatest structures in the universe?

The statistical characterization of the distribution of visible matter in the universe is a central problem in modern cosmology. In this respect, a crucial question still lacking a definitive answer concerns how large the greatest structures in the universe are. This point is closely related to whether or not such a distribution can be approximated as being homogeneous on large enough scales. Here we assess this problem by considering the size distribution of superclusters of galaxies and by leveraging the properties of Zipf–Mandelbrot law, providing a novel approach which complements the standard analysis based on the correlation functions. We find that galaxy superclusters are well described by a pure Zipf’s law with no deviations and this implies that all the catalogs currently available are not sufficiently large to spot a truncation in the power-law behavior. This finding provides evidence that structures larger than the greatest superclusters already observed are expected to be found when deeper redshift surveys will be completed. As a consequence, the scale beyond which galaxy distribution crossovers toward homogeneity, if any, should increase accordingly.

Bioclimatic Regularities of Change in the Density of Organic Carbon of the Steppe Soil in Different Regions of the World

The bioclimatic regularities between the average annual precipitation, average annual temperatures and the density of organic carbon in the soil layer of 0-30 cm of the steppes in the regions of the world are given. They are distinguished by a high certainty of quantization by asymmetric wave equations. It turned out that, due to the vibrational adaptation of organic carbon, precipitation and temperature are dependent on each other. For example, the model of the influence of precipitation on temperature includes the first term in the form of Laplace’s law (in mathematics), Mandelbrot’ law (in physics), Zipf-Perl (in biology), and Pareto (in econometrics). The second term is the biotechnical law of the author of the article, which gives the maximum change in the indicator. Both components form a trend that makes it possible to divide the precipitation interval into three stages: (1) with an increase in precipitation from 0 to 60 mm, the temperature decreases according to Mandelbrot’s law from 23.25 to 0.5 0С; (2) from 60 to 2100 mm, the temperature rises to 24 ° C; (3) with a further increase in precipitation over 2100 mm, a slow decrease in temperature occurs. The third term is an asymmetric wavelet with a constant half-period of 367.8 mm. A positive sign shows that in the steppes there is a positive oscillatory adaptation of temperature to changes in precipitation. In the interval of precipitation 0-350 mm, an oscillatory decrease in temperature occurs. It turns out that the first oscillation at 0 mm precipitation begins with a very high temperature gradient of thermal energy. The first interval includes Mongolia and Inner Mongolia. In the second interval of 350- 750 mm, an oscillatory increase in temperature occurs. Then, in the third interval 750-1050 mm, the temperature drops again. The second oscillation with a correlation coefficient of 0.9685 has clear precipitation boundaries in the range of 200-2000 mm. Due to the negative sign, the fluctuation is a crisis, inhibiting the rise in temperature. And the third fluctuation has a positive effect on the temperature. The mechanism of oscillatory adaptation in the steppe soil is so perfect that it changes for itself the conditions of the place where the grass grows. An amplitude-frequency analysis of each oscillation will make it possible to determine the specific particular effects of precipitation and temperature on each other and on the density of organic carbon. It was found that two-factor modeling of the change in the soil organic carbon density makes it possible to achieve an identification error even less than the absolute measurement error.

Inequalities of the Edmundson-Lah-Ribarč type for n-convex functions with applications

UDC 517.5 We derive some Edmundson – Lah – Ribarič type inequalities for positive linear functionals and -convex functions. Main results are applied to the generalized -divergence functional. Examples with Zipf – Mandelbrot law are used to illustrate the results.

A statistical test for correspondence of texts to the Zipf—Mandelbrot law

We analyse correspondence of a text to a simple probabilistic model. The model assumes that the words are selected independently from an infinite dictionary. The probability distribution correspond to the Zipf---Mandelbrot law. We count sequentially the numbers of different words in the text and get the process of the numbers of different words. Then we estimate Zipf---Mandelbrot law parameters using the same sequence and construct an estimate of the expectation of the number of different words in the text. Then we subtract the corresponding values of the estimate from the sequence and normalize along the coordinate axes, obtaining a random process on a segment from 0 to 1. We prove that this process (the empirical text bridge) converges weakly in the uniform metric on $C (0,1)$ to a centered Gaussian process with continuous a.s. paths. We develop and implement an algorithm for approximate calculation of eigenvalues of the covariance function of the limit Gaussian process, and then an algorithm for calculating the probability distribution of the integral of the square of this process. We use the algorithm to analyze uniformity of texts in English, French, Russian and Chinese.

Estimation of different entropies via Hermite interpolating polynomial using Jensen type functionals

We estimated the different entropies like Shannon entropy, Renyi divergences, Csiszar divergence by using the Jensen’s type functionals. The Zipf’s mandelbrot law and hybrid Zipf’s mandelbrot law are used to estimate the Shannon entropy. Further the Hermite interpolating polynomial is used to generalize the new inequalities for m-convex function.

Words ranking and Hirsch index for identifying the core of the hapaxes in political texts

This paper deals with a quantitative analysis of the content of official political speeches. We study a set of about one thousand talks pronounced by the US Presidents, ranging from Washington to Trump. In particular, we search for the relevance of the rare words, i.e. those said only once in each speech -- the so-called hapaxes. We implement a rank-size procedure of Zipf-Mandelbrot type for discussing the hapaxes' frequencies regularity over the overall set of speeches. Starting from the obtained rank-size law, we define and detect the core of the hapaxes set by means of a procedure based on an Hirsch index variant. We discuss the resulting list of words in the light of the overall US Presidents' speeches. We further show that this core of hapaxes itself can be well fitted through a Zipf-Mandelbrot law and that contains elements producing deviations at the low ranks between scatter plots and fitted curve -- the so-called king and vice-roy effect. Some socio-political insights are derived from the obtained findings about the US Presidents messages.

Estimation of different entropies via Lidstone polynomial using Jensen-type functionals

In this work, some new functional of Jensen-type inequalities are constructed using Shannon entropy, f-divergence, and Rényi divergence, and some estimates are obtained for these new functionals. Also using the Zipf–Mandelbrot law and hybrid Zipf–Mandelbrot law, we investigate some bounds for these new functionals. Furthermore, we generalize these new functionals for m-convex function using Lidstone polynomial.

Estimation of Different Entropies via Taylor One Point and Taylor Two Points Interpolations Using Jensen Type Functionals

In this work, we estimated the different entropies like Shannon entropy, Renyi divergences, Csiszar divergence by using the Jensen’s type functionals. The Zipf’s mandelbrot law and hybrid Zipf’s mandelbrot law are used to estimate the Shannon entropy. Further the Taylor one point and Taylor two points interpolations are used to generalize the new inequalities for m-convex function.

Estimation of different entropies via Abel–Gontscharoff Green functions and Fink’s identity using Jensen type functionals

In this work, we estimated the different entropies like Shannon entropy, Rényi divergences, Csiszár divergence by using Jensen’s type functionals. The Zipf’s–Mandelbrot law and hybrid Zipf’s–Mandelbrot law are used to estimate the Shannon entropy. The Abel–Gontscharoff Green functions and Fink’s Identity are used to construct new inequalities and generalized them for m-convex function.

New Estimations for Shannon and Zipf-Mandelbrot Entropies.

The main purpose of this paper is to find new estimations for the Shannon and Zipf–Mandelbrot entropies. We apply some refinements of the Jensen inequality to obtain different bounds for these entropies. Initially, we use a precise convex function in the refinement of the Jensen inequality and then tamper the weight and domain of the function to obtain general bounds for the Shannon entropy (SE). As particular cases of these general bounds, we derive some bounds for the Shannon entropy (SE) which are, in fact, the applications of some other well-known refinements of the Jensen inequality. Finally, we derive different estimations for the Zipf–Mandelbrot entropy (ZME) by using the new bounds of the Shannon entropy for the Zipf–Mandelbrot law (ZML). We also discuss particular cases and the bounds related to two different parametrics of the Zipf–Mandelbrot entropy. At the end of the paper we give some applications in linguistics.