发表机构
School of Mathematics and Statistics, Central South University; ESIME-Zacatenco, Instituto Politécnico Nacional（中南大学数学与统计学院; 国立理工学院萨卡特科分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对d-可求和曲线上的多超解析与元超解析函数空间，探究其基本黎曼型边值问题的可解性条件，相关成果可支撑分形复杂结构上的数学分析与偏微分方程研究。
AI 中文摘要
数学家阿夫龙·道格利斯（Avron Douglis）提出的超解析函数，是一类道格利斯代数值函数，其定义基于超复结构而非传统复分析中典型的标准柯西-黎曼方程。多超解析函数与元超解析函数类是道格利斯分析的高级推广，被用于偏微分方程与弹性力学的研究中，通过高阶迭代与非齐次项扩展了经典全纯函数的概念。本研究的目标是，针对定义在复平面上开、有界、单连通子集上的多超解析函数与元超解析函数空间，求解一类基本的黎曼型边值问题的可解性条件，该问题的边界仅需为闭的d-可求和曲线。在分形几何中，d-可求和性是用于定义分形区域边界的几何性质，支持在珍妮·哈里森（Jenny Harrison）与亚历克·诺顿（Alec Norton）定义的复杂结构上开展高级数学积分与微积分运算。
英文摘要
Hyperanalytic functions, in the sense established by the mathematician Avron Douglis, are Douglis algebra-valued functions defined via a hypercomplex structure rather than the standard Cauchy-Riemann equations characteristic of traditional complex analysis. The classes of polyhyperanalytic and meta-hyperanalytic functions represent advanced generalizations of Douglis's analysis. They are employed in the study of partial differential equations and elasticity, extending the concept of the classical holomorphic function through higher-order iterations and non-homogeneous terms. The aim of this work is to find solvability conditions for a fundamental Riemann-type boundary value problem for spaces of poly-hyperanalytic and meta-hyperanalytic functions defined on an open, bounded, simply connected subset of the complex plane, where the boundary need only be a closed d-summable curve. In fractal geometry, d-summability is a geometric property used to define the boundaries of fractal domains, enabling advanced mathematical integration and calculus on complex structures defined by Jenny Harrison and Alec Norton.
