The Euler-Glaisher Theorem over Totally Real Number Fields

In this paper, we study the partition theory over totally real number fields. Let $K$ be a totally real number field. A partition of a totally positive algebraic integer $\delta$ over $K$ is $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_r)$ for some totally positive integers $\lambda_i$ such that $\delta=\lambda_1+\lambda_2+\cdots+\lambda_r$. We find an identity to explain the number of partitions of $\delta$ whose parts do not belong to a given ideal $\mathfrak a$. We obtain a generalization of the Euler-Glaisher Theorem over totally real number fields as a corollary. We also prove that the number of solutions to the equation $\delta=x_1+2x_2+\cdots+nx_n$ with $x_i$ totally positive or $0$ is equal to that of chain partitions of $\delta$. A chain partition of $\delta$ is a partition $\lambda=(\lambda_1,\lambda_2,\ldots,\lambda_r)$ of $\delta$ such that $\lambda_{i+1}-\lambda_i$ is totally positive or $0$

The Sylvester Theorem and the Rogers-Ramanujan Identities over Totally Real Number Fields

In this paper, we prove two identities on the partition of a totally positive algebraic integer over a totally real number field which are the generalization of the Sylvester Theorem and that of the Rogers-Ramanujan Identities. Additionally, we give an another version of generalized Rogers-Ramanujan Identities

A Fuzzy-Based Multimedia Content Retrieval Method Using Mood Tags and Their Synonyms in Social Networks

The preferences of Web information purchasers are rapidly evolving. Cost-effectiveness is now becoming less regarded than cost-satisfaction, which emphasizes the purchaser’s psychological satisfaction. One method to improve a user’s cost-satisfaction in multimedia content retrieval is to utilize the mood inherent in multimedia items. An example of applications using this method is SNS (Social Network Services), which is based on folksonomy, but its applications encounter problems due to synonyms. In order to solve the problem of synonyms in our previous study, the mood of multimedia content is represented with arousal and valence (AV) in Thayer’s two-dimensional model as its internal tag. Although some problems of synonyms could now be solved, the retrieval performance of the previous study was less than that of a keyword-based method. In this paper, a new method that can solve the synonym problem is proposed, while simultaneously maintaining the same performance as the keyword-based approach. In the proposed method, a mood of multimedia content is represented with a fuzzy set of 12 moods of the Thayer model. For the analysis, the proposed method is compared with two methods, one based on AV value and the other based on keyword. The analysis results demonstrate that the proposed method is superior to the two methods