Dimensional Root Grand Unified Theory (DR-GUT): A Unified Theory for Root Operations in Arbitrary Dimensions维度根大统一理论
Fuente:
Zenodo
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Recurso digital |
| Pubblicazione: |
Zenodo
2026
|
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866902161026187264 |
|---|---|
| author | 郭, 洋洋 |
| author_facet | 郭, 洋洋 |
| contents | <p>根运算作为数学中最基础的运算之一,长期以来被认为是不同维度下相互独立的操作。平方根、立方根、四次方根等不同维度的根运算,在传统数学中被分别定义和计算,缺乏统一的理论框架。本文提出了维度根大统一理论(Dimensional Root Grand Unified Theory, DR-GUT),首次实现了对任意维度根运算的统一描述。我们证明,所有的根运算都遵循同一个通用的几何法则:体积增量沿超表面积线性分布。基于此,我们推导出了统一的近似公式 $$\sqrt[n]{a^n + \Delta} \approx a + \frac{\Delta}{n \cdot a^{n-1}}$$,该公式在小增量场景下具有极高的精度,且可以通过迭代扩展到任意增量大小。我们通过大量的数值实验验证了该理论的正确性,结果表明,该公式在所有维度下都具有极高的精度,相对误差均低于万分之一。该理论不仅统一了所有维度的根运算,还为数值计算、硬件设计、物理建模等领域提供了全新的工具。<br>As one of the most fundamental operations in mathematics, root operations have long been regarded as independent operations in different dimensions. Root operations of different dimensions, such as square root, cube root, and fourth root, are separately defined and computed in traditional mathematics, lacking a unified theoretical framework. In this paper, we propose the Dimensional Root Grand Unified Theory (DR-GUT), which for the first time achieves a unified description of root operations in arbitrary dimensions. We prove that all root operations follow the same universal geometric law: volume increment is linearly distributed along the hypersurface area. Based on this, we derive a unified approximation formula $$\sqrt[n]{a^n + \Delta} \approx a + \frac{\Delta}{n \cdot a^{n-1}}$$, which has extremely high accuracy in small increment scenarios and can be extended to arbitrary increment sizes through iteration. We verify the correctness of this theory through extensive numerical experiments, and the results show that this formula has extremely high accuracy in all dimensions, with relative errors all below one ten-thousandth. This theory not only unifies root operations of all dimensions, but also provides a brand new tool for fields such as numerical computation, hardware design, and physical modeling.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19990017 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Dimensional Root Grand Unified Theory (DR-GUT): A Unified Theory for Root Operations in Arbitrary Dimensions维度根大统一理论 郭, 洋洋 <p>根运算作为数学中最基础的运算之一,长期以来被认为是不同维度下相互独立的操作。平方根、立方根、四次方根等不同维度的根运算,在传统数学中被分别定义和计算,缺乏统一的理论框架。本文提出了维度根大统一理论(Dimensional Root Grand Unified Theory, DR-GUT),首次实现了对任意维度根运算的统一描述。我们证明,所有的根运算都遵循同一个通用的几何法则:体积增量沿超表面积线性分布。基于此,我们推导出了统一的近似公式 $$\sqrt[n]{a^n + \Delta} \approx a + \frac{\Delta}{n \cdot a^{n-1}}$$,该公式在小增量场景下具有极高的精度,且可以通过迭代扩展到任意增量大小。我们通过大量的数值实验验证了该理论的正确性,结果表明,该公式在所有维度下都具有极高的精度,相对误差均低于万分之一。该理论不仅统一了所有维度的根运算,还为数值计算、硬件设计、物理建模等领域提供了全新的工具。<br>As one of the most fundamental operations in mathematics, root operations have long been regarded as independent operations in different dimensions. Root operations of different dimensions, such as square root, cube root, and fourth root, are separately defined and computed in traditional mathematics, lacking a unified theoretical framework. In this paper, we propose the Dimensional Root Grand Unified Theory (DR-GUT), which for the first time achieves a unified description of root operations in arbitrary dimensions. We prove that all root operations follow the same universal geometric law: volume increment is linearly distributed along the hypersurface area. Based on this, we derive a unified approximation formula $$\sqrt[n]{a^n + \Delta} \approx a + \frac{\Delta}{n \cdot a^{n-1}}$$, which has extremely high accuracy in small increment scenarios and can be extended to arbitrary increment sizes through iteration. We verify the correctness of this theory through extensive numerical experiments, and the results show that this formula has extremely high accuracy in all dimensions, with relative errors all below one ten-thousandth. This theory not only unifies root operations of all dimensions, but also provides a brand new tool for fields such as numerical computation, hardware design, and physical modeling.</p> |
| title | Dimensional Root Grand Unified Theory (DR-GUT): A Unified Theory for Root Operations in Arbitrary Dimensions维度根大统一理论 |
| url | https://doi.org/10.5281/zenodo.19990017 |