Measuring the homogeneity or diversity of a distribution is critical in many fields, from evaluating the code produced by a cryptographic algorithm to the biodiversity as a key indicator of ecosystem health. Different measures of homogeneity are necessary to accommodate different evaluation views in different fields and applications, in the same way as we can extract different means of a same sequence. Shannon entropy, introduced to measure the content of information of a distribution, and one-parameter entropies such as Tsallis and Rényi ones, have been frequently used as homegeneity measures. In this paper, we review first general properties of an homogeneity measure, and then, based on two different approaches by Cichocki and Amari and by Topsøe respectively, we consider entropies derived from f-divergences.

Natural capital defines planetary boundaries and provides a basis for sustainable development. This study reviews previous theoretical developments and confirms that natural capital accounting within the Inclusive Wealth (IW) framework provides a robust link between current capital assets and intergenerational well-being. This study contributes to the literature by combining theoretical advances with practical applications to address criticisms of empirical practice and improve the reliability and scope of cross-country natural capital accounting. An analysis of natural capital levels and changes in 163 economies over the past 30 years reveals significant regional disparities in the decline of global natural capital. In low-income countries, consumption driven by population growth and primary production patterns is severely depleting renewable natural capital. In middle-income countries, urbanization exacerbates natural capital depletion by substituting other forms of capital for natural capital. The wealth status of major G20 economies points to intensive environmental costs and loss of ecosystem services under technological progress, which ignores public ecosystem externalities. This study demonstrates the urgency of natural capital depletion awareness in the management of all economies and highlights the ability of natural capital accounting within the IW framework to inform policy decisions on sustainable growth.

Measuring a country’s sustainable development by its gross domestic product (GDP) is insufficient to capture the loss of capital that determines future human welfare. To address this, we propose the inclusive wealth index, which integrates biophysical quantities and monetary values of natural, human, and produced capital. We analyzed the level of Inclusive Wealth in 163 countries over the past 30 years to assess sustainable development goals (SDGs) going beyond GDP. Global wealth has suffered significant losses in natural capital, with the biased accumulation of capital assets leading to unsustainable and unequal development. In low-income countries, soaring population levels and biased capital investments exacerbate the depletion of natural capital. Our results underscore the critical role of natural capital and inclusive capital management in sustainable development. Policymakers can use this information to make capital investments in their economies and promote recovery from COVID-19 that aligns with the SDGs, the Paris Climate Agreement, and initiatives beyond GDP.

Cities play a crucial role in regional sustainable development through trade linkages with surrounding economies. This study extends the inclusive wealth (IW) conceptual framework for footprint analysis, offering a comprehensive production–consumption perspective to measure regional sustainability. We empirically analyse the IW footprint for 1880 municipal-level economies in Japan by using their territorial IW accounting. We measure sustainability in the hierarchical value chains across cities and prefectures. Our findings suggest the unsustainability of production and consumption across cities in Japan, as the biased wealth clustering in cross-prefecture value chains led to wealth inequality. Additionally, we observe the distorted natural and human capital utilization characterized as the general depreciation of natural capital and shortage of human capital, which arise by the value chain participation. Our results underscore the importance of capital management and regulation in value chains. Sustainable development policy interventions must focus on optimizing inclusive capital asset management to maintain a non-declining level of wealth. This research unveils the intricate relationship between cities and their surroundings, providing valuable insights for policymakers aiming to enhance regional sustainability.

Abstract Human capital is widely acknowledged as an inclusive resource that encompasses the knowledge and expertise of citizens, which propels economic development. This research integrates education data with demographic, health, and socioeconomic factors to construct a cross-country human capital monetary accounting framework. We evaluate the levels of educational attainment and the total adult population by stage of life expectancy. Subsequently, we assess education returns based on lifetime income and educational attainment. This accounting tracks the level and changes of human capital in 166 countries from 1990 to 2020, providing multiple indicators to measure human capital for sustainability. The primary findings indicate that globally, the growth trajectory of human capital has declined over the past decade, suggesting a threshold for human capital growth under current investment levels. Based on these findings, it is recommended that policymakers consider this shift and align long-term human capital investment with sustainable growth paths.

The impact of international trade on national decarbonization outcomes is inconclusive. Taking Germany as a case study, we developed a relative intersectoral linkage analysis based on the global value chain (GVC) framework. We identified nations' emission changes due to the transition of composition, scale, and technology in the fragmented trade network. The results show that in the process of reducing emissions, on the one hand, the supply of local emission-intensive sectors to the domestic economy and international trade has shrunk. On the other hand, we found comprehensive emission intensity reduction driven by downstream sectors' production, even if there was carbon leakage in the participation of backward GVCs. However, when emission-intensive production was concentrated in the local market, GVC backward participation positively affected the emission intensity of the local economy. According to the findings, we assert that continuous technological innovation and reconstruction of the value chain are necessary for sustainable decarbonization from the local to the global economy.

In this paper, we present order invariance theoretical results for weighted quasi-arithmetic means of a monotonic series of numbers. The quasi-arithmetic mean, or Kolmogorov–Nagumo mean, generalizes the classical mean and appears in many disciplines, from information theory to physics, from economics to traffic flow. Stochastic orders are defined on weights (or equivalently, discrete probability distributions). They were introduced to study risk in economics and decision theory, and recently have found utility in Monte Carlo techniques and in image processing. We show in this paper that, if two distributions of weights are ordered under first stochastic order, then for any monotonic series of numbers their weighted quasi-arithmetic means share the same order. This means for instance that arithmetic and harmonic mean for two different distributions of weights always have to be aligned if the weights are stochastically ordered, this is, either both means increase or both decrease. We explore the invariance properties when convex (concave) functions define both the quasi-arithmetic mean and the series of numbers, we show its relationship with increasing concave order and increasing convex order, and we observe the important role played by a new defined mirror property of stochastic orders. We also give some applications to entropy and cross-entropy and present an example of multiple importance sampling Monte Carlo technique that illustrates the usefulness and transversality of our approach. Invariance theorems are useful when a system is represented by a set of quasi-arithmetic means and we want to change the distribution of weights so that all means evolve in the same direction.

Frequency histograms are ubiquitous, being practically used in any field of science. In this paper, we present a partial order for frequency histograms and, to our knowledge, no order of this kind has been yet defined. This order is based on the stochastic order of discrete probability distributions and it has invariance properties that make it unique. First, we model a frequency histogram as a sequence of bins associated with a discrete probability (or relative frequency) distribution. Then, we consider that two histograms are ordered if they are defined on the same sequence of bins and their respective frequency distributions are stochastically ordered. The ordering can be easily spotted because the respective cumulative distribution functions of the frequencies of two ordered histograms do not cross each other. Finally, with each bin we can associate a representative value of the bin, and for two ordered histograms it holds that all quasi-arithmetic means (such as arithmetic, harmonic, and geometric mean) of the representative values weighted by the frequencies are ordered in the same direction than the histograms are. Our theoretical study is supported by three experiments in the fields of image processing, traffic flow, and income distribution.

Information theory, and the concept of information channel, allows us to calculate the mutual information between the source (input) and the receiver (output), both represented by probability distributions over their possible states. In this paper, we use the theory behind the information channel to provide an enhanced interpretation to a Social Accounting Matrix (SAM), a square matrix whose columns and rows present the expenditure and receipt accounts of economic actors. Under our interpretation, the SAM's coefficients, which, conceptually, can be viewed as a Markov chain, can be interpreted as an information channel, allowing us to optimize the desired level of aggregation within the SAM. In addition, the developed information measures can describe accurately the evolution of a SAM over time. Interpreting the SAM matrix as an ergodic chain could show the effect of a shock on the economy after several periods or economic cycles. Under our new framework, finding the power limit of the matrix allows one to check (and confirm) whether the matrix is well-constructed (irreducible and aperiodic), and obtain new optimization functions to balance the SAM matrix. In addition to the theory, we also provide two empirical examples that support our channel concept and help to understand the associated measures.

In 1999, the Grain to Green Program (GTGP) was initiated in China as payment of ecosystem service and the largest land conservation program. This paper evaluates the impact of this conservation program on the regional economic structural change in Guizhou Province. We generate the agriculture-based regional input–output tables of 2002, 2007 and 2012. The production and employment inducements are calculated by using the Leontief inverse matrix. The key sectors are figured out by applying the linkage analysis. The RAS analysis is used for categorizing sectors as the potentially growing or shrinking industry. The results indicate the undesired effect of the GTGP and the EMP on the upstream supplier and downstream industries for agricultural production. Rural household expenditure was changed by the subsidies and shows negative effect on agricultural production. In the second stage of policy implementation, we found the depression of rural economies. The shrinking potential of the rural sector forced the further cutback of these upstream and downstream industries. The surplus rural labors find difficulties in searching for jobs in the local labor market and should move out of this region. From these results, we suggest the local government should reconsider the economic influence of the GTGP and EMP rather than emphasize the environmental issues and intend to improve rural welfare.

This paper examines intra-provincial market integration in Guizhou Province, China. We measure spatial market integration by utilizing time series data of rice and soybean prices. Results indicate long run rice market integration exists in three regional markets in eight recent years. However, the rice-surplus market does not dominate rice distribution, and this market is integrated with other deficit markets with longer time lags. Long run soybean market integration exists only between transports hub and the nearest market. A remote market is separated from soybean distribution in other markets. From this result, we conclude that the intra-provincial market is inefficient for local farmers to distribute their products. In order to increase the local farmers’ incomes, more flexible incentive polices need to be adopted to improve farm gate market efficiency.

Leaves of vascular plants in tropical and subtropical rain forests are frequently colonized by epiphylls. Most epiphylls are lichens and liverworts,but a few algae,cyan-bacteria and minute invertebrate animals also can occur. The composition of epiphyll communities is a sensitive indicator of seasonal changes of forest environments and microenvironment heterogeneity,and it can directly reflect habitat factors including moisture,temperature and illumination. Furthermore,epiphylls play significant roles in the formation and maintenance of biodiversity,as well as in the circulation of nutrients and water in ecosystems. In addition,they are indicators of forest responses to global changes. Epiphylls obtain nutrients independently. However,some experiments have provided evidence that substances can be exchanged between epiphylls and their host plants. Although epiphylls cover host leaf surface,whether the shading affects photosynthesis of the host plant is controversial. It has been hypothesized recently that leaves can photo-acclimate to the cover of the epiphyll community and photosynthesis of covered leaf areas can be compensated fully. The functions of epiphylls in nutrient and water circulation of forest ecosystems have attracted more attention in recent years. Nitrogen fixed by the epiphyll community provides 10%–25% of the nitrogen for the understory of tropical forest ecosystems. Epiphylls can also retain atmospheric moisture to alleviate drought in the dry season. There may be evolutionary balances between epiphyllous organisms and their host plants,and the relationships are likely influenced by mul-tiple,poorly understood factors. More research is needed on relationships between epiphylls and their hosts.

In order to make clear the relationships between photosynthesis and leaf N, leaf P and SLA of tropical trees, and test the differences in the relationships among life-form groups (trees, shrub-like trees and shrubs), seedlings and saplings of 101 species from a tropical montane rain forest, located in the Diaoluo Mountain of Hainan Island, were selected. The net photosynthesis based on area and mass (A area and A mass), leaf nitrogen content based on area and mass (N area and N mass), leaf phosphorus content based on area and mass (P area and P mass) and specific leaf area (SLA) were measured and/or calculated. The results showed that A area and A mass tended to follow the order of shrubs > trees > shrub-like trees. One-way ANOVA showed that the difference in A area between shrubs and shrub-like trees was significant (p<0.05), and for A mass there were significant differences between shrubs and shrub-like trees and between shrubs and tree species (p<0.05). The relationships between A area and N mass were highly significant in all three life-form groups and for all species (p<0.0001). The correlation between A area and P mass was highly significant in shrubs (p = 0.0038), shrub-like trees (p < 0.0002) and for all species (p<0.0001), but not significant in trees (p>0.05). The relationship between A area and SLAwas highly significant in shrubs (p = 0.0006), trees (p<0.0001) and for all species (p<0.0001), however this relation was not significant in shrub-like trees (p>0.05). The relationships between A mass and leaf N and SLA were highly significant in all three life-form groups and for all species (p<0.0001). For A mass and leaf P, there were significant correlations in tree groups (p = 0.0377) and highly significant correlations in shrub groups (p = 0.0004), shrub-like tree groups (p = 0.0018) and for all species (p < 0.0001). Stepwise regression showed that predicted A mass values were closer to the observed values than those for predicted A area values. Thus, it can be concluded that the relationships obtained from seedling and sapling measurements are close to those from mature individuals; correlations between photosynthesis and N mass, P mass and SLA traits are significant and the relationships are stronger and more stable for A mass than for A area.

We used inter-simple sequence repeat (ISSR) markers to investigate the genetic structure of Eurya emarginated Makine for four populations collected from three islands in the Zhoushan Archipelago, and the result was then used for genetic diversity comparison between Eurya emarginated Makine, Ilex integra and Machilus thunbergii. The average values of percentage of polymorphic loci is 37.95%, effective number of alleles(Ne=1.223), Nei’s genetic diversity (H=0.132) and Shannon’s information index (I=0.200) indicated that the Eurya population had lower genetic variations than both the Ilex integra and Machilus thunbergii species. The genetic differentiation for populations of Eurya emarginated Makine(Gst=0.202)is lower than those of Ilex integra and Machilus thunbergii. UPGMA cluster for individuals revealed that the isolation between the islands could not completely obstruct the genetic exchanges between different populations of Eurya emarginated Makine, and the excursion of plant seeds floating depressed the value of genetic differentiation, while the exchanges of Ilex integra and Machilus thunbergii populations were greatly isolated by the islands.

Genetic diversity and differentiation of populations of Thuidium cymbifolium between 3 planted spruce stands(PSS) and 3 naturally regenerated stands(NRS) were compared using random amplified polymorphic DNA(RAPD) marker.For PSS and NRS,the average percentage of polymorphic loci(PPL=12.7%,10.0%),Nei's genetic diversity(HE = 0.042,0.027) and Shannon's information index(S=0.064,0.043) were revealed by POPGENE,respectively.The results of population differentiation among the populations from PSS(Gst=0.592,Fst=0.639) and NRS(Gst=0.702,Fst= 0.695),together with the results of UPGMA cluster analysis and of principal components analysis showed that gene flow between populations was limited.AMOVA revealed a significantly higher level of population genetic diversity existed in PSS than that in NRS,which indicated that artificial reforestation after forest clearcut was effective in promoting the recovery of genetic diversity of the recolonized moss species.