Set Theory and Logic

2508 Submissions

[2] viXra:2508.0078 [pdf] submitted on 2025-08-12 20:28:43

Iterative Expansion of Generalized Mapping Theory and Description of Taylor Series Expansion

Authors: Hongtao Ling
Comments: 5 Pages.

Based on the quadruple framework (object set A, operation set F, result set B, and generation relation ⊢) proposed in the Generalized Mapping Theory [1], this paper makes a key extension to address the limitations of its recursive mechanism, constructing an enhanced generalized mapping framework for dynamic mathematical process modeling. Inspired by the ideas of "explicit operation set" in Chapter 2 and "multi-stage process modeling" in Chapter 4 of the preprint, this study innovatively introduces a state transfer mechanism to solve the core problem of original function information loss in Taylor series expansion. By modeling the Taylor expansion as a recursive sequence of operations, the new framework achieves three breakthroughs: first, it explicitly separates the derivative calculation rules from the state transfer path, extending the "behavior mathematization" concept of the preprint to the field of pure mathematics; second, it establishes a state-driven recursive paradigm that completely retains the function itself and expansion point parameters. Using Taylor expansion as a paradigmatic case, this study verifies the universal descriptive ability of the enhanced generalized mapping for continuous mathematical processes, providing a mathematical tool with both operation traceability and structural expressiveness for dynamic systems such as numerical solutions of differential equations and function approximation, further realizing the theoretical vision of "establishing a complete path of behavior-result generation relations" proposed in the preprint.
Category: Set Theory and Logic

[1] viXra:2508.0074 [pdf] submitted on 2025-08-11 19:05:13

Tensor Extension of Generalized Mapping and Conversion Between Its Two Forms

Authors: Hongtao Ling
Comments: 5 Pages.

Traditional generalized function theories (such as Schwartz's distribution theory and Colombeau algebra) have limitations in describing certain phenomena [1]. To address this issue, the Generalized Mapping theory is proposed, which realizes the mathematical modeling of dynamic behavior-result correlations through a four-element framework consisting of an object set A, an operation set F, a result set B, and a generative relation ⊢. However, when the Generalized Mapping theory was initially proposed, its two forms were formulated and defined using sets. Although set-based definitions can describe the basic logic of the theory, they are insufficient in practical applications, especially in computer simulations. Since vector, matrix, and even tensor computations are frequently required (despite sets being occasionally treatable as vectors), it is necessary to theoretically extend the Generalized Mapping theory to tensors. Additionally, there are scenarios where conversion between its two forms is needed. Therefore, this paper presents the tensor extension of the Generalized Mapping theory and discusses the theoretical conditions required for converting between its two forms.
Category: Set Theory and Logic