Algebraic-analytical Study of non-local pseudo-differential equations in complex domains

复域非局部伪微分方程的代数解析研究

• 批准号: 15540155
• 负责人: ISHIMURA Ryuichi
• 金额: $ 2.11万
• 依托单位: Chiba University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540155/
• 关键词: Algebraic analysis; Differential-difference equations; Pseudo-differential equations; Differential equations of infinite ordre; partial differential equations; Convolution equations; Microlocal analysis of sheaves; Cauchy problem

The aims of this research were as follows :[1]To study algebraic-analytically non-local pseudo-differential equations in complex domains.[2]To construct holomorphic solutions to such equations and to establish the operational calculus.[3]To study algebraic-analytically differential-differnce eqations in complex domains.For [1], at first, we have defined cohomologically the classes of the non-local pseudo-differential operators operating to holomorphic functions defined at a fixed point, as well as their composition and the operation to functions. And secondly, using the Fourier-Sato transform, we re-defined the class of non-local pseudo-differential operators and proved functorially their composition. For [2], in the case of constant coefficient, we established a concrete integral representation formula for solutions to the non-local pseudo-differential equation, using the formal inverse operator. We also obtained concretely generators of the solution space of corresponding homogeneous equation. For [3], using the formula obtained in [2], we calculated special solutions to differential-difference equations with constant coefficient.

本文的主要研究内容如下：[1]研究复域上的代数解析非局部伪微分方程。[2]To构造这些方程的全纯解并建立运算演算。[3]To研究复域上的代数解析微分差分方程.在文[1]中，我们首先定义了在不动点上作用于全纯函数的非局部伪微分算子类，以及它们的合成和对函数的作用.其次，利用Fourier-Sato变换，重新定义了一类非局部伪微分算子，并泛函地证明了它们的复合性。对于文献[2]，在常系数的情况下，利用形式逆算子建立了非局部伪微分方程解的具体积分表示公式.具体地得到了相应齐次方程解空间的生成元。对于文[3]，我们利用文[2]中得到的公式计算了常系数微分差分方程的特解。

项目成果

Masaru Nagisa: "Single generation and rank of C*-algebras""Operator Algebras and Applications", Advanced Study in Pure Mathematics, Mathematical Society of Japan. 38. 135-143 (2004)

Masaru Nagisa：“C*-代数的单代和秩”“算子代数及其应用”，纯数学高级研究，日本数学会。

Almost automorphic solutions for abstract functional differential equations

• DOI: 10.1016/s0022-247x(03)00531-6
• 发表时间: 2003-10
• 期刊: Journal of Mathematical Analysis and Applications
• 影响因子: 1.3
• 作者: Y. Hino;S. Murakami

On the exact WKB analysis of operators admitting infinitely many phases.

关于允许无限多个相的算子的精确 WKB 分析。

• 发表时间: 2004
• 期刊: Adv. in Math. 181
• 作者: T.Ogawa;Y.Taniuchi;M.Ozawa;T.Aoki
• 通讯作者: T.Aoki

Some relations for multiple zeta values and connection formulas for the Gauss hypergeometric functions

多个zeta值的一些关系和高斯超几何函数的连接公式

• 期刊: Publications RIMS, Kyouto University (to appear)
• 作者: AOKI Takashi;OHNO Yasuo
• 通讯作者: OHNO Yasuo

Yoshiyuki Hino, Satoru Murakami: "Almost automorphic solutions for abstract functional differential equations"Journal of Mathematical Analysis and Applications. 286. 741-752 (2003)

Yoshiyuki Hino、Satoru Murakami：“抽象泛函微分方程的几乎自守解”数学分析与应用杂志。