Entropy of measure-preserving or continuous actions of amenable discrete groups allows for various equivalent approaches. Among them are those given by the techniques developed by Ollagnier and Pinchon on the one hand and the Ornstein–Weiss lemma on the other. We extend these two approaches to the context of actions of amenable topological groups. In contrast to the discrete setting, our results reveal a remarkable difference between the two concepts of entropy in the realm of non-discrete groups: while the first quantity collapses to 0 in the non-discrete case, the second yields a well-behaved invariant for amenable unimodular groups. Concerning the latter, we moreover study the corresponding notion of topological pressure, prove a Goodwyn-type theorem, and establish the equivalence with the uniform lattice approach (for locally compact groups admitting a uniform lattice). Our study elaborates on a version of the Ornstein–Weiss lemma due to Gromov.

This study aims to explore the dependencies on the cryptocurrency market using social network tools. We focus on the correlations observed in the cryptocurrency returns. Based on the sample of cryptocurrencies listed between January 2015 and December 2022 we examine which cryptos are central to the overall market and how often major players change. Static network analysis based on the whole sample shows that the network consists of several communities strongly connected and central, as well as a few that are disconnected and peripheral. Such a structure of the network implies high systemic risk. The day-by-day snapshots show that the network evolves rapidly. We construct the ranking of major cryptos based on centrality measures utilizing the TOPSIS method. We find that when single measures are considered, Bitcoin seems to have lost its first-mover advantage in late 2016. However, in the overall ranking, it still appears among the top positions. The collapse of any of the cryptocurrencies from the top of the rankings poses a serious threat to the entire market.

A series of papers by Hickey (1982, 1983, 1984) presents a stochastic ordering based on randomness. This paper extends the results by introducing a novel methodology to derive models that preserve stochastic ordering based on randomness. We achieve this by presenting a new family of pseudometric spaces based on a majorization property. This class of pseudometrics provides a new methodology for deriving the randomness measure of a random variable. Using this, the paper introduces the Gini randomness measure and states its essential properties. We demonstrate that the proposed measure has certain advantages over entropy measures. The measure satisfies the value validity property, provides an adequate extension to continuous random variables, and is often more appropriate (based on sensitivity) than entropy in various scenarios.

For a class of volume-preserving partially hyperbolic diffeomorphisms (or non-uniformly Anosov) $f\colon {\mathbb {T}}^d\rightarrow {\mathbb {T}}^d$ homotopic to linear Anosov automorphism, we show that the sum of the positive (negative) Lyapunov exponents of f is bounded above (respectively below) by the sum of the positive (respectively negative) Lyapunov exponents of its linearization. We show this for some classes of derived from Anosov (DA) and non-uniformly hyperbolic systems with dominated splitting, in particular for examples described by Bonatti and Viana [SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115(1) (2000), 157–193]. The results in this paper address a flexibility program by Bochi, Katok and Rodriguez Hertz [Flexibility of Lyapunov exponents. Ergod. Th. & Dynam. Sys. 42(2) (2022), 554–591].

Political polarization has become an increasingly salient issue worldwide, but a systematic examination of the variation and sources of mass polarization across countries is limited by current measurement methods. This work proposes a nonparametric, entropy-based measure of mass political polarization. It exploits the specific structure of ordinal distributions in public opinion data, makes no prior assumptions about the form and spacing of the data, and can still draw reliable measures of issue-based polarization. We demonstrate the theoretical and practical superiority of the measure with analytical comparisons and simulations. We then apply the proposed measure to questions about mass polarization in the USA, the relationship between radical parties and polarization in Europe, and cross-country trends in affective and ideological polarization.

Cation-exchange equilibrium for Ca-K-montmorillonite was studied at 35°, 50°, and 90°C and at three total normalities of the equilibrium solution (0.1, 0.05, and 0.01 N). Changes of the standard free energy for the exchange from K-montmorillonite to Ca-montmorillonite were determined to be −53, −270, and −393 cal/eq at 35°, 50°, and 90°C, respectively. Changes of the standard enthalpy and entropy were 1.7 kcal/eq and 5.6 cal/eq/degree at 35°C, respectively. The sign of the change of the standard free energy was found to be determined mainly by the entropy change, in particular, by the hydration entropy of the cations.

The calculation of the excess functions indicates that the mixing model of Ca-K-montmorillonite approximates that of a regular solution. Montmorillonite having potassium equivalent ion fraction of 0.1 to 0.7 consists of a random interstratification of Ca-montmorillonite (15.6 Å) and K-montmorillonite (12.6 Å).

Ion-exchange experiments in expanding clay minerals conducted over a wide range of surface ionic compositions and ionic strength produce variable mass-action selectivity coefficients. When the exchanging ions are of unequal charge, tactoid structure appears to influence selectivity, although configurational entropy of adsorbed ions may also generate variable selectivity. The degree of deviation from ideal mass-action exchange is related to the dissimilarity of the ions undergoing exchange. Data involving trivalent ion adsorption on smectites suggest that mass-action is a poor approximation when the adsorbing and desorbing ions have different hydration energies and charge. No form of exchange equation is successful in describing ion exchange for a wide range of experimental conditions, although the fluctuation of the selectivity coefficient follows consistent trends with changing experimental conditions. The strong adsorption of high-charge ions on clays is not exothermic, but must be driven by the increasing disorder of ions and/or water.

This paper investigates the separation property in binary phase-segregation processes modelled by Cahn-Hilliard type equations with constant mobility, singular entropy densities and different particle interactions. Under general assumptions on the entropy potential, we prove the strict separation property in both two and three-space dimensions. Namely, in 2D, we notably extend the minimal assumptions on the potential adopted so far in the literature, by only requiring a mild growth condition of its first derivative near the singular points $\pm 1$, without any pointwise additional assumption on its second derivative. For all cases, we provide a compact proof using De Giorgi’s iterations. In 3D, we also extend the validity of the asymptotic strict separation property to the case of fractional Cahn-Hilliard equation, as well as show the validity of the separation when the initial datum is close to an ‘energy minimizer’. Our framework offers insights into statistical factors like particle interactions, entropy choices and correlations governing separation, with broad applicability.

We define a notion of substitution on colored binary trees that we call substreetution. We show that a point fixed by a substreetution may (or not) be almost periodic, and thus the closure of the orbit under the $\mathbb {F}_{2}^{+}$-action may (or not) be minimal. We study one special example: we show that it belongs to the minimal case and that the number of preimages in the minimal set increases just exponentially fast, whereas it could be expected a super-exponential growth. We also give examples of periodic trees without invariant measures on their orbit. We use our construction to get quasi-periodic colored tilings of the hyperbolic disk.

The exchange of Na, K, and Ca ions in vermiculite has been studied between 50 and 150°C at a total normality of 0.1. The free energy changes were negative in the Na → K, Ca → K, and Na → Ca exchange reactions, and the cation preference in the vermiculite was Na < Ca < K in the temperature range examined. The enthalpy changes, calculated by application of the van't Hoff equation to the relation between equilibrium constant and temperature, were positive. The cation preference in the vermiculite seems to have been largely controlled by the increase in entropy which was due to the positive values of the entropy change term in the solid for the Na-K and Ca-K exchanges and the positive values of the entropy change in the solution for the Na-Ca exchange. From a thermodynamic analysis of the cation-mixing properties in vermiculite, the observed free energy change in the Na-K exchange was determined solely by the differences of the intrinsic electrostatic binding energy of the cations on the clay surface. That in the Na-Ca exchange was interpreted by taking into account an extra interaction energy of Ca-Ca pairs in addition to the intrinsic energy change. Furthermore, an additional interaction energy between K ions and the clay surface appears to have been added to the free energy change in the Ca-K exchange. In the Na → K and Ca → K exchange reactions a structural modification occurred in the K-equivalent fraction (X̄K) range 0.05–0.6, and a regularly interstratified phase of 15-Å and 10-Å members was formed at 0.05 < X̄K < 0.4.

We prove that if two free probability-measure-preserving (p.m.p.) ${\mathbb Z}$-actions are Shannon orbit equivalent, then they have the same entropy. The argument also applies more generally to yield the same conclusion for free p.m.p. actions of finitely generated virtually Abelian groups. Together with the isomorphism theorems of Ornstein and Ornstein–Weiss and the entropy invariance results of Austin and Kerr–Li in the non-virtually-cyclic setting, this shows that two Bernoulli actions of any non-locally-finite countably infinite amenable group are Shannon orbit equivalent if and only if they are measure conjugate. We also show, at the opposite end of the stochastic spectrum, that every ${\mathbb Z}$-odometer is Shannon orbit equivalent to the universal ${\mathbb Z}$-odometer.

Adsorption isotherms for water on homoionic (Ca2+, Li+, Na+, and K+) exchanged forms of Crook County, Wyoming, montmorillonite (CMS Source Clay SWy-1) were measured between 25° and 70°C using a vacuum microbalance having automated control of water vapor pressure. From these adsorption data, integral enthalpies and entropies of adsorption were calculated. Both quantities were negative, but decreased in magnitude with increasing amounts of adsorbed water. For all four cationic forms of the clay, the amount of initial water adsorption at 25°C and at low relative humidities was sensitive to the sample temperature during prior evacuation of water, less water being adsorbed by samples evacuated at 100°C compared with samples evacuated at 25°. For Ca- and Na-montmorillonite, these changes were reversible after several subsequent desorption/adsorption cycles, but recovery was not observed for the Li-SWy-1 clay, probably because of migration of Li+ into the aluminosilicate structure.

The heat capacities of kaolinite (7 to 380 K) and of dimethyl sulfoxide (DMSO) intercalated kaolinite (20 to 310 K) were measured by adiabatically shielded calorimetry. The third law entropy of kaolinite, S298, is 200.9 ± 0.5 J·mol−1K−1.

The “melting point” of the DMSO in the intercalate, 288.0 ± 0.2 K, is 3.7 K lower than that of pure DMSO, 291.67 K. The heat capacity of DMSO in the intercalate above 290 K is approximately 5.2 J·mol−1·K−1 smaller than that of pure liquid DMSO at the same temperature.

Elastin function is to endow vertebrate tissues with elasticity so that they can adapt to local mechanical constraints. The hydrophobicity and insolubility of the mature elastin polymer have hampered studies of its molecular organisation and structure-elasticity relationships. Nevertheless, a growing number of studies from a broad range of disciplines have provided invaluable insights, and several structural models of elastin have been proposed. However, many questions remain regarding how the primary sequence of elastin (and the soluble precursor tropoelastin) governs the molecular structure, its organisation into a polymeric network, and the mechanical properties of the resulting material. The elasticity of elastin is known to be largely entropic in origin, a property that is understood to arise from both its disordered molecular structure and its hydrophobic character. Despite a high degree of hydrophobicity, elastin does not form compact, water-excluding domains and remains highly disordered. However, elastin contains both stable and labile secondary structure elements. Current models of elastin structure and function are drawn from data collected on tropoelastin and on elastin-like peptides (ELPs) but at the tissue level, elasticity is only achieved after polymerisation of the mature elastin. In tissues, the reticulation of tropoelastin chains in water defines the polymer elastin that bears elasticity. Similarly, ELPs require polymerisation to become elastic. There is considerable interest in elastin especially in the biomaterials and cosmetic fields where ELPs are widely used. This review aims to provide an up-to-date survey of/perspective on current knowledge about the interplay between elastin structure, solvation, and entropic elasticity.