The Lamé class of Lorenz curves

Sarabia, José Marı́a; Jordá, Vanesa; Trueba, Carmen

doi:10.1080/03610926.2013.775306

Cited by 14 publications

(4 citation statements)

References 42 publications

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“…One other metric in economy is the Gini index to quantify inequalities based on Lorenz curves. In many cases this can be modeled efficiently with Lamé curves or superellipses [34,35]. Recently, it was shown that Gini and Theil indices from economy can also be used in the study of natural phenomena.…”

Section: Allometry Laws Of Physics and Production Functionsmentioning

confidence: 99%

Conquering Mount Improbable

Gielis

2023

Proceedings of the 1st International Symposium on Square Bamboos and the Geometree (ISSBG 2022)

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Entropy is originally a concept from thermodynamics that distinguishes between useful and useless energy 1 . This led to the Second Law of Thermodynamics, which states that entropy always increases. The Hamiltonian principle states that the differential equation for the Second Law is equivalent to the integral equation for Least Action.In the information age, entropy has been found to be related to complexity. There are two main approaches, one of which is based on C. Shannon (1916Shannon ( -2001 and the other on A. N. Kolmogorov (1903N. Kolmogorov ( -1987. The former is communication theory, while the latter is the basis of Algorithmic Information Theory (AIT), which studies the shortest algorithm for encoding a message that yields the "best possible compression". This is of course also usage-based and refers to 'optimal' methods of data processing. The commonly used example is a particular sequence of binary digits, symbols or characters. Each of these sequences is called a message. Two sequences (or strings) are:1 "Useful energy" is somewhat anthropocentric, in the sense of "usefulness for humans".Our scientific and technological worldviews are largely dominated by the concepts of entropy and complexity. Originating in 19th-century thermodynamics, the concept of entropy merged with information in the last century, leading to definitions of entropy and complexity by Kolmogorov, Shannon and others. In its simplest form, this worldview is an application of the normal rules of arithmetic. In this worldview, when tossing a coin, a million heads or tails in a row is theoretically possible, but impossible in practice and in real life. On this basis, the impossible (in the binary case, the outermost entries of Pascal's triangle 𝑥𝑥𝑥𝑥 𝑛𝑛𝑛𝑛 and 𝑦𝑦𝑦𝑦 𝑛𝑛𝑛𝑛 for large values of 𝑛𝑛𝑛𝑛) can be safely neglected, and one can concentrate fully on what is common and what conforms to the law of large numbers, in fields ranging from physics to sociology and everything in between. However, in recent decades it has been shown that what is most improbable tends to be the rule in nature. Indeed, if one combines the outermost entries 𝑥𝑥𝑥𝑥 𝑛𝑛𝑛𝑛 and 𝑦𝑦𝑦𝑦 𝑛𝑛𝑛𝑛 with the normal rules of arithmetic, either addition or multiplication, one obtains Lamé curves and power laws respectively. In this article, some of these correspondences are highlighted, leading to a double conclusion. First, Gabriel Lamé's geometric footprint in mathematics and the sciences is enormous. Second, conic sections are at the core once more. Whereas mathematics so far has been exclusively the language of patterns in the sciences, the door is opened for mathematics to also become the language of the individual. The probabilistic worldview and Lamé's footprint can be seen as dual methods. In this context, it is to be expected that the notions of information, complexity, simplicity and redundancy benefit from this different viewpoint.

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Section: Allometry Laws Of Physics and Production Functionsmentioning

confidence: 99%

Conquering Mount Improbable

Gielis

2023

Proceedings of the 1st International Symposium on Square Bamboos and the Geometree (ISSBG 2022)

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“…Известно, что модель (13) хорошо зарекомендовала себя при изучении неравенства доходов населения (Sarabia, Jordá and Trueba, 2013).…”

Section: модель кривой паретоunclassified

Распределение Размеров Штатов, Округов И Городов США: Новая Информация О Форме Неравенства

Grachev¹

2020

Preprint

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В статье предлагается новый подход к исследованию закономерностей распределения размеров в системах расселения, основанный на анализе формы кривой Парето (PC). Для изучения формы PC использованы коэффициент Джини, асимметрии и, по аналогии с физикой фазовых переходов, показатель степени PC в окрестности нуля. Выполнен эмпирический анализ PC различных уровней агрегации системы расселения США. Форму распределения размеров штатов исследовали по десятилетиям с 1790 по 2010 год. Пространственный анализ формы PC округов и городов выполнен для 2010 года. Результаты проведенного эмпирического исследования показали, что PC штатов на протяжении 220 лет имела левостороннюю асимметрию. PC округов и городов имела как правостороннюю, так и левостороннюю асимметрию. Полученные результаты объясняют, в каких случаях распределение Парето, имеющее PC с правосторонней асимметрией, и логнормальное распределение с симметричной PC объективно могут не соответствовать реальным системам расселения. В качестве альтернативы степенному и логнормальному распределению рассмотрена аналитически простая двухпараметрическая модель с широким диапазоном асимметрии PC, объединяющая в себе свойства комбинации степенного и логнормального распределений. Верификация модели показала, что она адекватно описывает размеры поселений однородных систем расселения.

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“…It is known that the model(13) has shown good results in the study of inequality of incomes of the population (Sarabia, Jordá and Trueba, 2013).…”

Section: The Pareto Curve Modelmentioning

confidence: 99%

Size Distribution of States, Counties, and Cities in the USA: New Inequality Form Information

Grachev

2020

Preprint

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In this article, we propose a new approach for studying the patterns of size distribution in settlement systems, based on the analysis of the shape of the Pareto curve (PC). To study the shape of the PC, we used the Gini coefficient, the asymmetry coefficient, and, by analogy with the physics of phase transitions, critical exponent — the index of the PC degree in the neighborhood of zero. An empirical analysis of the PC of various levels of aggregation in the US settlement system has been performed. The form of size distribution of states was studied by decades from 1790 to 2010. The spatial analysis of the PC shape for counties and cities was performed for 2010. The results of an empirical study showed that the PC of the states had left-hand asymmetry over 220 years. The PC of districts and cities had both right-hand and left-hand asymmetries. The obtained results explain in which cases the Pareto distribution having a PC with right-hand asymmetry, and the lognormal distribution with a symmetric PC may not correspond objectively to real settlement systems. As an alternative to power-series distribution and lognormal distribution, we considered an analytically simple two-parameter model with a wide range of PC asymmetry that combines the properties of power-series distribution and lognormal distribution. Verification of the model showed that it adequately described the size of settlements in homogeneous settlement systems.

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The Lamé class of Lorenz curves

Cited by 14 publications

References 42 publications

Conquering Mount Improbable

Conquering Mount Improbable

Распределение Размеров Штатов, Округов И Городов США: Новая Информация О Форме Неравенства

Size Distribution of States, Counties, and Cities in the USA: New Inequality Form Information

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