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Study on Multifractal and its Applications to the Digital Sum Problems

多重分形及其在数字和问题中的应用研究

基本信息

批准号：
12640135
负责人：
OKADA Tatsuya
金额：
$ 0.45万
依托单位：
Fukushima Medical University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640135/
关键词：
Fractal Multifractal Digital Sum Problems Multinomial Measure 多項測度

项目摘要

The purpose of this research project is to study a system of infinitely many difference equations and functional equation with respect to the multinomial measure, which is a typical multifractal measure, and to apply this study to the digital sum problems expanded in the p-adic number. We investigated the digital sum problems systematic by using the multinomial measure. Each investigator considered his subject positively and made usefulcontribution. We enumerate the results in the following.1. Usual digital sum problems were solved by using the multinomial measures.2. We introduced a generalization of the power and the exponential sums, which contained information per digit, and gave explicit formulas of them by using the multinomial measure.3. As an application of the formula obtained above, we gave an explicit formula of the number of occurrences of subblock in the p-adic expansion.4. By generalizing the distribution function of multinomial measure with complex coefficients, we gave another explicit formula of Coquet's summation formula related to the binary digits in the multiple of three.Above complexity is only one instance of a generalization of the multinomial measure. In the process of this study, we found that new classes of measures, which contain the multinomial measures, are still more effective for the digital sum problems. We should prepare an effective theory of these measures for applications of the digital sum problems.

本课题的目的是研究关于多项式测度的无穷多个差分方程和函数方程组，多项式测度是一种典型的多重分形测度，并将这一研究应用于p-adic数上展开的数字和问题。利用多项式测度系统地研究了数字和问题。每位研究者都对自己的课题给予了积极的评价，并做出了有益的贡献。我们列举的结果如下。1.利用多项式测度解决了一般的数字和问题.引入了幂和和指数和的推广，每一位都包含信息，并利用多项式测度给出了它们的显式表达式.作为上述公式的应用，我们给出了p-adic展开式中子块出现次数的显式公式.通过推广复系数多项式测度的分布函数，给出了与三的倍数的二进制数有关的Coquet求和公式的另一个显式公式，上述复杂性只是多项式测度的推广的一个例子.在研究的过程中，我们发现包含多项式测度的新测度类对于数字和问题仍然是更有效的。我们应该准备一个有效的理论，这些措施的数字和问题的应用。