This study has considered the problem of finding best PROXIMITY PAIR in fuzzy metric spaces and uniformly convex fuzzy Banach spaces for fuzzy cyclic contraction map. We prove the uniqueness of this point in uniformly fuzzy Banach spaces. We also give an algorithm to find a best PROXIMITY point for the map S in setting of a uniformly convex fuzzy Banach spaces.

A new geometric notion on a nonempty and convex PAIR of subsets of a convex metric space X, called semi-normal structure, is introduced and used to investigate the existence of best PROXIMITY PAIRs for a new class of mappings, called strongly noncyclic relatively C-nonexpansive. We also study the structure of minimal sets of strongly noncyclic relatively C-nonexpansive mappings in the setting of convex metric spaces.

This paper is concerned with the best PROXIMITY PAIR problem in Hilbert spaces. Given two subsets A and B of a Hilbert space H and the set-valued maps F: A® 2B and G: A0®2A0, where A0= {xÎ A: ||x - y||=d (A, B) for some yÎ B}, best PROXIMITY PAIR theorems provide sufficient conditions that ensure the existence of anx0 2 A such thatd(G (x0), F (x0)) =d (A, B).

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 this article, we consider PAIR frames in Banach spaces and introduce Banach PAIR frames. Some various concepts in the frame theory such as frames, Schauder frames, Banach frames and atomic decompositions are considered as special kinds of (Banach) PAIR frames. Some frame-like inequalities for (Banach) PAIR frames are presented. The elements that participant in the construction of (Banach) PAIR frames are characterized. It is shown that a Banach space X has a Banach PAIR frame with respect to a Banach scalar sequence space `, when it is precisely isomorphic to a complemented subspace of `. It is shown that if we are allowed to choose the scalar sequence space, PAIR frames and Banach PAIR frames with respect to the chosen scalar sequence space denote the same concept.

In this article, we introduce a new notion of proximal contraction, named as generalized S-proximal contraction and derive a common best PROXIMITY point theorem for proximally commuting non-self mappings, thereby yielding the common optimal APPROXIMATE solution of some  xed point equations when there is no common solution. We furnish illustrative examples to highlight our results. We extend some results existing in the literature.

Typographical data entry errors and incomplete documents, produce imperfect records in real world databases. These errors generate distinct records which belong to the same entity. The aim of APPROXIMATE Record Matching is to find multiple records which belong to an entity. In this paper, an algorithm for APPROXIMATE Record Matching is proposed that can be adapted automatically with input error patterns. In field matching phase, edit distance method is used. Naturally, it had been customized for Persian language problems such as similarity of Persian characters, usual typographical errors in Persian, etc. In record matching phase, the importance of each field can be determined by specifying a coefficient related to each field. Coefficient of each field must be dynamically changed, because of changes of typographical error patterns. For this reason, Genetic Algorithm (GA) is used for supervised learning of coefficient values. The simulation results show the high abilities of this algorithm compared with other methods (such as Decision Trees).

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.