Mean-Value Formulae for the Neighbourhood of the Typical Cell of a Random Tessellation

1994 ◽  
Vol 26 (3) ◽  
pp. 565-576 ◽  
Author(s):  
S. N. Chiu
Keyword(s):  
Maximum Entropy ◽  
Mean Value ◽  
Entropy Theory ◽  
Simulation Studies ◽  
Exact Form ◽  
Neighbouring Cell ◽  
Higher Dimensional ◽  
Typical Cell ◽  
The Mean ◽  
Maximum Entropy Theory

The mean number of edges of a randomly chosen neighbouring cell of the typical cell in a planar stationary tessellation, under the condition that it has n edges, has been studied by physicists for more than 20 years. Experiments and simulation studies led empirically to the so-called Aboav's law. This law now plays a central role in Rivier's (1993) maximum entropy theory of statistical crystallography. Using Mecke's (1980) Palm method, an exact form of Aboav's law is derived. Results in higher-dimensional cases are also discussed.

Mean-Value Formulae for the Neighbourhood of the Typical Cell of a Random Tessellation

1994 ◽  
Vol 26 (03) ◽  
pp. 565-576 ◽  
Author(s):  
S. N. Chiu
Keyword(s):  
Maximum Entropy ◽  
Mean Value ◽  
Entropy Theory ◽  
Simulation Studies ◽  
Exact Form ◽  
Neighbouring Cell ◽  
Higher Dimensional ◽  
Typical Cell ◽  
The Mean ◽  
Maximum Entropy Theory

The mean number of edges of a randomly chosen neighbouring cell of the typical cell in a planar stationary tessellation, under the condition that it has n edges, has been studied by physicists for more than 20 years. Experiments and simulation studies led empirically to the so-called Aboav's law. This law now plays a central role in Rivier's (1993) maximum entropy theory of statistical crystallography. Using Mecke's (1980) Palm method, an exact form of Aboav's law is derived. Results in higher-dimensional cases are also discussed.

Maximum entropy theory

1985 ◽  
Vol 41 (2) ◽  
pp. 113-122 ◽  
Author(s):  
A. K. Livesey ◽  
J. Skilling
Keyword(s):  
Maximum Entropy ◽  
Entropy Theory ◽  
Maximum Entropy Theory

scholarly journals Land use, macroecology, and the accuracy of the Maximum Entropy Theory of Ecology: A case study of Azorean arthropods

2021 ◽  
Author(s):  
Micah Brush ◽  
Thomas J. Matthews ◽  
Paulo A.V. Borges ◽  
John Harte
Keyword(s):  
Land Use ◽  
Land Use Change ◽  
Metabolic Rate ◽  
Maximum Entropy ◽  
Management Practices ◽  
Entropy Theory ◽  
State Variables ◽  
Arthropod Communities ◽  
Natural Pasture ◽  
Maximum Entropy Theory

AbstractHuman activity and land management practices, in particular land use change, have resulted in the global loss of biodiversity. These types of disturbances affect the shape of macroecological patterns, and analyzing these patterns can provide insights into how ecosystems are affected by land use change. The Maximum Entropy Theory of Ecology (METE) simultaneously predicts many of these patterns using a set of ecological state variables: the number of species, the number of individuals, and the total metabolic rate. The theory’s predictions have been shown to be successful across habitats and taxa in undisturbed natural ecosystems, although previous tests of METE in relation to disturbance have focused primarily on systems where the state variables are changing relatively quickly. Here, we assess predictions of METE applied to a different type of disturbance: land use change. We use METE to simultaneously predict the species abundance distribution (SAD), the metabolic rate distribution of individuals (MRDI), and the species–area relationship (SAR) and compare these predictions to arthropod data from 96 sites at Terceira Island in the Azores archipelago across four different land uses of increasing management intensity: 1. native forest, 2. exotic forest, 3. semi-natural pasture, and 4. intensive pasture. Across these patterns, we find that the forest habitats are the best fit by METE predictions, while the semi-natural pasture consistently provided the worst fit. The intensive pasture is intermediately well fit for the SAD and MRDI, and comparatively well fit for the SAR, though the residuals are not normally distributed. The direction of failure of the METE predictions at the pasture sites is likely due to the hyperdominance of introduced spider species present there. We hypothesize that the particularly poor fit for the semi-natural pasture is due to the mix of arthropod communities out of equilibrium and the changing management practices throughout the year, leading to greater heterogeneity in composition and complex dynamics that violate METE’s assumption of static state variables. The comparative better fit for the intensive pasture could then result from more homogeneous arthropod communities that are well adapted to intensive management, and thus whose state variables are less in flux.

scholarly journals Derivations of the Core Functions of the Maximum Entropy Theory of Ecology

Entropy ◽  
2019 ◽  
Vol 21 (7) ◽  
pp. 712 ◽  
Author(s):  
Alexander Brummer ◽  
Erica Newman
Keyword(s):  
Species Richness ◽  
Structure Function ◽  
Maximum Entropy ◽  
Information Entropy ◽  
Lagrange Multipliers ◽  
Metabolic Energy ◽  
Entropy Theory ◽  
State Variables ◽  
The Core ◽  
Maximum Entropy Theory

The Maximum Entropy Theory of Ecology (METE), is a theoretical framework of macroecology that makes a variety of realistic ecological predictions about how species richness, abundance of species, metabolic rate distributions, and spatial aggregation of species interrelate in a given region. In the METE framework, “ecological state variables” (representing total area, total species richness, total abundance, and total metabolic energy) describe macroecological properties of an ecosystem. METE incorporates these state variables into constraints on underlying probability distributions. The method of Lagrange multipliers and maximization of information entropy (MaxEnt) lead to predicted functional forms of distributions of interest. We demonstrate how information entropy is maximized for the general case of a distribution, which has empirical information that provides constraints on the overall predictions. We then show how METE’s two core functions are derived. These functions, called the “Spatial Structure Function” and the “Ecosystem Structure Function” are the core pieces of the theory, from which all the predictions of METE follow (including the Species Area Relationship, the Species Abundance Distribution, and various metabolic distributions). Primarily, we consider the discrete distributions predicted by METE. We also explore the parameter space defined by the METE’s state variables and Lagrange multipliers. We aim to provide a comprehensive resource for ecologists who want to understand the derivations and assumptions of the basic mathematical structure of METE.

Study on Probabilistic Damage Constitutive Relation of Rocks Based on Maximum-Entropy Theory

2020 ◽  
Vol 20 (2) ◽  
pp. 06019018 ◽  
Author(s):  
Bin Hu ◽  
Peng-Zhi Pan ◽  
Wei-Wei Ji ◽  
Shuting Miao ◽  
Decai Zhao ◽  
...  
Keyword(s):  
Maximum Entropy ◽  
Constitutive Relation ◽  
Entropy Theory ◽  
Maximum Entropy Theory

MEAN VALUE PROPERTY OF HARMONIC FUNCTION ON THE HIGHER-DIMENSIONAL SIERPINSKI GASKET

Fractals ◽  
2020 ◽  
Vol 28 (05) ◽  
pp. 2050077
Author(s):  
YIPENG WU ◽  
ZHILONG CHEN ◽  
XIA ZHANG ◽  
XUDONG ZHAO
Keyword(s):  
Harmonic Function ◽  
Harmonic Functions ◽  
Sierpinski Gasket ◽  
Mean Value ◽  
Sierpiński Gasket ◽  
Mean Value Property ◽  
Average Value ◽  
Higher Dimensional ◽  
The Mean

Harmonic functions possess the mean value property, that is, the value of the function at any point is equal to the average value of the function in a domain that contain this point. It is a very attractive problem to look for analogous results in the fractal context. In this paper, we establish a similar results of the mean value property for the harmonic functions on the higher-dimensional Sierpinski gasket.

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