A hereditary class on a set X is a nonempty collection of subsets of X which is closed under subsets. In this paper, we present a new structure of Proximity spaces by using a hereditary class, called H-Proximity spaces, as a generalization of Efremoviˇ, c Proximity spaces, I-Proximity spaces and coarse Proximity spaces. Some properties of this Proximity structure and generalized topology induced by it are studied.

Countries are adjacent or opposite in physical, political, economic, social, cultural, and scientific dimensions. Proximity or distance among countries in terms of the above dimensions can cause or prevent strong inter-country scientific collaboration. The objectives of this research are primarily to identify the dimensions that better explain the intensity of international scientific collaboration, and secondly, to investigate the fluctuations in the intensity of the influence of each dimension over time. To achieve the above-mentioned aims, and based on the integration of Proximity literature with the CAGE distance framework, the determining factors of international scientific collaboration were extracted. The population of the research consisted of 167 independent countries in the period of 2002-2017. In this research, the network of co-author relations between countries is considered as an indicator of international scientific collaboration, whose data was collected from the Web of Science database. Also, data on Proximity dimensions were collected from various databases (United Nations, International Trade Center, etc. ). This research has an explanatory nature and is based on the analysis of secondary data with a quantitative approach (a gravity model). Model effects were estimated using R package fixest. The application of this model showed that the intensity of international scientific collaboration has a positive relationship with Proximity and a negative relationship with the distance between countries in different dimensions. The findings obtained from the comparison of the explanatory power of the Proximity dimensions indicated the high explanatory power of social Proximity. Nevertheless, the findings of the model improvement showed that the simultaneous presence of political, economic, and social dimensions also have an explanatory power close to that of the social Proximity. Examining the fluctuations in the influence of Proximity dimensions over time showed a significant increase in the influence of Proximity in economic, social, and scientific dimensions, a common language in the cultural dimension, intra-regional collaboration, and collaboration between neighboring countries in the physical dimension. Meanwhile, the influence of geographical distance in the physical dimension, colonial relations in the political dimension, and religious and ethnic commonalities in the cultural dimension have decreased significantly. It is only in the political dimension that the effect of most factors is partial (such as the difference in terms of the degree of realization of the liberal democratic government or the efficiency of the government, and the imposition of sanctions against the other party) and sometimes is fixed (such as common membership in international agreements).

In this paper we introduce geometric contraction map and give a new condition for the existence and uniqueness of best Proximity point of geometric contractions. We also consider the convergence of iterates to Proximity points in metric spaces.

In the neoclassical and evolutionary economics, the emphasis is placed on the importance of Proximity to facilitate the innovation process. Neoclassical economics with the focus on the aggregation advantages, such as cost savings and economy of scale and the evolutionary economics with the emphasis on the benefits of firms’ Proximity such as knowledge spillovers, emply the necessity of designing policies and tools based on the geographical Proximity. In the geographical Proximity structures, teams, technological firms, startups and industrial companies are located together. The teams and companies located in these structures, while having a joint infrastructure at a lower cost, benefit from the knowledge and technology spillovers. In this chapter, the importance of geographical Proximity in innovation is presented from different theories’ perspective, and then various structures based on geographical Proximity, including accelerator, incubator, science and technology park, science and technology town, technology town and industrial estate commensurate with different stages of the firms’ life cycle, are reviewed and compared. Finally, an example of aggregation of these structures in a geographical region, i. e. Isfahan province has been presented.

Some grammarians and philologists believe that some words with reduced endings (majrur) are not so on the basis of the rules applying to words with reduced ending; i.e., neither there is any of the letters of reduction prior to them nor have they been posed as annexed or as subordinate to a word with reduced ending.Thereby, such a word should have diacritics other than jarr but is read as majrur (with reduced ending), and the reason for this diacritical mark is its Proximity to a majrur word.Due to the belief in the authenticity of Proximity, some scholars have accordingly justified certain readings (qara’at) of the Holy Qur'an.In the present discussion, some instances in which Proximity is claimed are dealt with and the scholars' views are studied and critically reViewed. As it will be clarified during the discussion, what the believers in Proximity say does not seem to be so much as what they have imagined to be.