These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin- Zelevinsky’s cluster algebras. Then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generators of cluster algebras. From quiver representations, we proceed to the cluster category, which yields a complete categorification of the cluster algebra and its combinatorial underpinnings.

Construction industry is one of the most important industries in the world that has a significant role in the development of countries. Therefore, in the recent decade, many countries accomplish extensive researches for growing their share in an international market. One of the economic approaches, which considered for this reason in several industries in the recent years, is the industrial cluster. However, because of the unique nature of construction industry, many dimensions of this strategy have not been clarified within construction industry. One of these dimensions, that have important role in a comprehensive perception of industrial cluster approach in the construction industry, is the construction cluster map.In this article, a comprehensive map for the construction cluster has been established by the use of grounded theory that illustrates vividly, the actors and activity that contribute to construction industry.

Background: Congenital Adrenal Hyperplasia (CAH) is an inherited hereditary autosomal recessive heredity, which is often induced by mutation in the CYP21A2 gene. The aim of the present study was to determine the prevalence of cluster exon 6 mutation among patients with CAH deficient in 21-OHD enzyme in Iranian population.Materials and Methods: In the current descriptive study, blood samples were collected from 25 patients with CAH who referred to Ali Asghar Hospital in Tehran. After extraction of genomic DNA, the region containing the cluster gene exon 6 gene was amplified using PCR and then sequenced. Finally, by analyzing sequences, the frequency of cluster mutation exon 6 was determined in the population studied.Results: A total of 15 patients (60%) had a cluster mutation of exon 6, so that in all of them, three mutations V236E, I236N, and M238K were observed as heterozygote. This mutation in the healthy parents of these patients was also identified as heterozygote.Conclusion: It seems that cluster E6 mutation in the heterozygote form alone does not lead to disease, and it is necessary to examine the presence of other mutations in the gene. This mutation may be present as compound heterozygote along with other mutations in the rest of the exons in patients.

One of the shortcomings of the existing clustering methods is their problems dealing with different shape and size clusters. On the other hand, most of these method sare designed for especial cluster types or have good performance dealing with particular size and shape of clusters. The main problem in this connection is how to define a dissimilarity criterion to make this algorithm capable of clustering general data, which include clusters of different shapes and sizes. Another important objective that must be considered is the computational complexity of any new algorithms.In this paper a new approach to fuzzy clustering is proposed in which a model for each cluster is estimated during learning.Gradually besides, dissimilarity metric for each cluster is defined, updated and used for the next step. In our approach, instead of associating a single cluster type to each cluster, we assume a set of possible cluster types for each cluster with different grades of possibility. Then, a truncation which can be expressed as an attention mechanism focuses on the most probable cluster types for each cluster. This selection step subsides the computational load dramatically while speeds up the clustering.The proposed clustering method which has the capability to deal with partial labeled data is implemented on two families of data, first in presence of partially labeled data, then with fully unlabeled data. Comparing the experimental results of this method with several important existing algorithms, demonstrates the superior performance of proposed method. The merit of this method is its ability to deal with clusters of different shape and size while it computes a fuzzy membership value to different shapes for each cluster.

In this paper, relations among the membership values of generalized fuzzy languages such as intuitionistic fuzzy language, interval-valued fuzzy language and vague language are studied. It will aid in studying the properties of one language when the properties of another are known. Further, existence of a minimized finite automaton with vague (final) states for any vague regular language recognized by a finite automaton with vague final) states is shown in this paper. Finally, an efficient algorithm is given for minimizing the finite automaton with vague (final) states. Similarly, it can be shown for intuitionistic fuzzy regular language. These may contribute to a better understanding of the role of finite automaton with vague (final) states or the finite automaton with intuitionistic fuzzy (final) states while studying lexical analysis, decision making etc.