Gaussian integer is one of basic algebraic integers. In this article we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaussian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rat...

We report on the result of investigating which relationship is important among hypernym and hyponym relationships in retrieval keyword expansion. Moreover, we report the effect of the keyword expansion and the contents complement for spoken document retrieval for SCR lecture retrieval task and SCR passage retrieval task. Spoken Document Retrieval by contents complement and keyword expansion usi...

The purpose of this article is to modify Ishikawa iterative process to have strong convergence for an infinite family nonexpansive mappings. Convergence theorems are established in a real Banach space. The results presented in this paper mainly improve the corresponding results announced in [3], [15] and [31].

J. Takada, M. Hoshi, T. Nagatomo1 , M. Yamamoto, T. Imanaka , T. Takatsuji4, I. Yoshikawa4, S. Endo5, M. Ishikawa, B. I. Gusev6, and A. K. Sakerbaev6 Research Institute for Radiation Biology and Medicine, Hiroshima University, Hiroshima 734-8553, Japan 1 Nara University of Education, Nara 630-8528, Japan, Kanazawa University, Ishikawa 923-1224, Japan Kyoto University, Osaka 590-0494, Japan, 4Na...

Graduate School of Science (T.L.S., T.S., I.H.-N.) and Graduate School of Agriculture (T.L.S., Y.T.), Kyoto University, Sakyo-ku, Kyoto 606–8502, Japan; Graduate School of Biological Sciences, Nara Institute of Science and Technology, Ikoma 630–0101, Japan (M.F., Y.F.); Research Institute for Bioresources and Biotechnology, Ishikawa Prefectural University, Ishikawa 921–8836, Japan (M.M.); RIKEN...

The purpose of this paper is, to establish, by extensive use of the minor summation formula of pfaffians exploited in (Ishikawa, Okada, and Wakayama, J. Algebra 183, 193 216) certain new generating functions involving Schur polynomials which have a product representation. This generating function gives an extension of the Littlewood formula. During the course of the proof we develop some techni...

We prove the equivalence of the convergence of the Mann and Ishikawa iterations with errors for uniformly continuous generalized Φ-pseudocontractive mappings in normed linear spaces. Our results extend and improve the corresponding results of Xu, 1998, Kim et al., 2009, Ofoedu, 2006, Chidume and Zegeye, 2004, Chidume, 2001, Chang et al. 2002, Liu, 1995, Hirano and huang, 2003, C. E. Chidume and...