• Journal of the Chungcheong Mathematical Society
• /
• v.21 no.4
• /
• pp.467-470
• /
• 2008

Abstract. We show that the cusp 1=2 is a Weierstrass point of ${\Gamma}_1(4p)$ if p is a prime greater than 7.

• Journal of applied mathematics & informatics
• /
• v.27 no.3_4
• /
• pp.737-748
• /
• 2009

Using the notion of intuitionistic fuzzy subgroups, its roughness is discussed. With respect to a congruence relation on a group, several properties about the lower and upper approximations of a subset of a group are investigated.

• Journal of the Chungcheong Mathematical Society
• /
• v.24 no.2
• /
• pp.345-349
• /
• 2011

In this work, we find the genera of arithmetic subgroups $\langle{\Gamma}_{\Delta}(N),{\Phi}\rangle$ of $GL_2^+(\mathbb{R})$ generated by congruence subgroup ${\Gamma}_{\Delta}(N)$ and the Fricke involution $\Phi$.

• Bulletin of the Korean Mathematical Society
• /
• v.46 no.5
• /
• pp.823-833
• /
• 2009

It is well-known that if a convex hyperbolic polygon is constructed as a fundamental domain for a subgroup of the modular group, then its translates by the group elements form a locally finite tessellation and its side-pairing transformations form a system of generators for the group. Such hyperbolically convex polygons can be obtained by using Dirichlet's and Ford's polygon constructions. Another method of obtaining a fundamental domain for subgroups of the modular group is through the use of a right coset decomposition and we call such domains standard fundamental domains. In this paper we give subgroups of the modular group which do not have hyperbolically convex standard fundamental domain containing only inequivalent cusps.

• Journal of Intelligence and Information Systems
• /
• v.17 no.1
• /
• pp.17-35
• /
• 2011

The collective intelligence is a common-based production by the collaboration and competition of many peer individuals. In other words, it is the aggregation of individual intelligence to lead the wisdom of crowd. Recently, the utilization of the collective intelligence has become one of the emerging research areas, since it has been adopted as an important principle of web 2.0 to aim openness, sharing and participation. This paper introduces an approach to seek the collective intelligence by cognition of the relation and interaction among individual participants. It describes a methodology well-suited to evaluate individual intelligence in information retrieval and classification as an application field. The research investigates how to derive and represent such cognitive intelligence from individuals through the application of fuzzy relational theory to personal construct theory and knowledge grid technique. Crucial to this research is to implement formally and process interpretatively the cognitive knowledge of participants who makes the mutual relation and social interaction. What is needed is a technique to analyze cognitive intelligence structure in the form of Hasse diagram, which is an instantiation of this perceptive intelligence of human beings. The search for the collective intelligence requires a theory of similarity to deal with underlying problems; clustering of social subgroups of individuals through identification of individual intelligence and commonality among intelligence and then elicitation of collective intelligence to aggregate the congruence or sharing of all the participants of the entire group. Unlike standard approaches to similarity based on statistical techniques, the method presented employs a theory of fuzzy relational products with the related computational procedures to cover issues of similarity and dissimilarity.