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Osaka Journal of Mathematics is published quarterly by the joint editorship of the Department of Mathematics, Graduate School of Science, Osaka University, and the Department of Mathematics, Faculty of Science, Osaka City University and the Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University with the cooperation of the Department of Mathematical Sciences, Faculty of Engineering Science, Osaka University. The Journal is devoted entirely to the publication of original works in pure and applied mathematics.