This paper gives yet another proof of the Pythagorean theorem and also of Dijkstra\'s generalization.

The approach angle method of reaching law which is based on conventional reaching law has the advantage of reducing the reaching time by incorporating large approach angle. However the disadvantage of this is that it will demand larger control effort and when applied to discrete systems, will lead to wider quasi sliding mode band (QSMB). A small approach angle while delaying the reaching will fetch a narrow QSMB. In this context, this paper presents a boundary layer (BL) based optimal sliding mode control for fast reaching with reduced QSMB which resorts to two different approach angles; one that serve to increase reaching speed up to a BL in the vicinity of the user designated sliding surface and the other to reduce QSMB. The proposed approach resorts to optimal control in the reaching phase by tracking the point on the sliding surface that can be reached in minimum time. This point is the one that can be reached with an angle of by the approach anglereaching law. Simulation results are presented to substantiate the advantages associated with the proposed methodology.

The present study aimed to analyze the national solid waste policy in Brazil, focusing on urban sustainability from the perspective of Local Development, exploring the relationship between the management and management of urban solid waste, since its generation, collection, and final disposal of waste consumer leftovers, produced by public and private companies or by individuals, this is regulated by Law 12.305/2010, also known as the National Solid Waste Policy (PNRS) of Brasil. This law was created to guide the actions of governments and society. In addition to the PNRS, the concern with environmental issues is also present in the Federal Constitution of 1988, also incorporating the structure of other laws that complement each other. The study has a qualitative approach, with bibliometric technical procedures and content analysis, based on the database of the Coordination for the Improvement of Higher Education Personnel (CAPES), covering the period from 2010 to 2020. The textual body analyzed was composed of 60 articles and instrumentalized using the Iramuteq software, which made it possible to read the equivalent of 10.003 words (some of the words that make up the abstracts of the selected scientific articles). Among the results, it was found that integrated management and solid waste management between public, private, and civil society can assist in the effective search for improvements in environmental balance, generating income, and ensuring urban sustainability.

This work aims to compare the Human Development Index (HDI) and sustainable of emancipated Municipalities with those of the respective Municipalities in the State of Santa Catarina, seeking to verify whether there was an advance or a setback resulting from the emancipatory process. The research is characterized by the quantitative, documentary and bibliographic approach. In the collection of data, data on the municipalities of Santa Catarina that have undergone the emancipation process since 1995 were searched. Data on the installation of the municipalities were collected, as well as the HDI indexes, populations and elements of their characterization on websites and documents. . At the end of the research, it was observed in all cases that there were no setbacks. It was found that in some cases there was an expressive development index in relation to the host cities in a sustainable way.

This note attempts to re-estimate the income inequality in China with the data provided in Li et al. (2020). We find that their estimates of the income distribution with the “mixed data” that combines household survey and the data for the top income groups underestimated the top income inequality in China. The estimates of this note with the same “mixed data” shows that the distribution of income inequality in China is significantly different from that of Li et al. (2020) and in particular the shares of the top incomes are heavily concentrated in the people such as the top 1% or top 0.1%. In addition, we show that the estimates of the top income inequality with the “mixed data” appears rather similar to the estimates of the top wealth inequality in the existing literature.