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.

‎Let $(C,D)$ be a nonempty pair of disjoint subsets of a metric space. ‎Main purpose of this paper is to present a range of a convergence sequence to $u\in C\cup D$ such that $d(Tu,fu)=dist(C,D)$‎, for mappings $T,f:C\cup D\to C\cup D$. ‎In fact, ‎we give a generalization of best proximity point results for cyclic contractive mappings. ‎To this end‎, ‎we consider an example is presented to support the main result. ‎‎