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摘要/Abstract： 四色定理是平面图着色领域经典难题，现有公认证明依赖计算机大规模构型枚举，缺少纯粹人工拓扑推导的严谨证明。本文以平面拓扑为基础，提出将任意连通平面图逐层分解为嵌套环域结构的剖分方法；利用同胚变换消去二度冗余顶点以保持面邻接关系不变，分层构建环域染色规则：相邻嵌套环层依据层数奇偶分配两套基底色实现层间正常着色，单个环域内部采用二染色完成子区域配色。全局两层基底色搭配单层两套局部色构成四色配色系统。通过双层奇数分片嵌套环域算例验证该染色方案的完备性，最终从拓扑构造角度给出四色定理完整初等证明，全程无需计算机辅助枚举。

关键词/Keywords：四色定理；平面图；环域剖分；面着色；同胚变换

摘要/Abstract： 基于人工智能（ＡＩ）的想象力，探索人工智能发展的终极场景，是一个超越人类思维或想象力且具有说服力的有效途径；通过这种途径而导出的人工智能发展的终极结论－－AI确实能帮我们人类突破一部分想象力，但ＡＩ终究会进化到一个连它自己都无法向我们解释清楚的境界--虽然看不懂，但却是一切奇迹的起点。
Leveraging the imagination of Artificial Intelligence (AI) to explore its ultimate developmental scenarios offers a compelling and effective pathway that transcends the boundaries of human thought. The ultimate conclusion derived through this approach is that while AI can indeed help humanity break through certain imaginative constraints, it will eventually evolve into a realm so profound that it cannot be explained even by itself. Though it may remain incomprehensible to us, it serves as the genesis of all future miracles.

关键词/Keywords： 人工智能（ＡＩ），想象力，认知论，人类进化，文明发展，物理规律。
Artificial Intelligence (AI), Imagination, Epistemology, Human Evolution, Civilizational Development, Laws of Physics.

波利尼亚克猜想：数学表达式的一个推导/Polignac's conjecture: Derivation of a Mathematical Formula cnxiv  cnXiv2024080101P      作者/Author： 胡坚/Jian Hu     发布于: 2024

Abstract: With using the segmented Eratosthenes' sieve approach, a definite mathematical formula describing Polignac's conjecture, p_(k+1)=p_k+2h, is deduced from a newly uncovered unusual distribution of prime numbers in the p_k~l_k plot (l_k is the segment length and p_k the prime number), where the parameter h is a positive integer unique to prime numbers. Validity of the said formula is proved by mathematical induction.

Keywords: number theory, Polignac's conjecture, sieve of Eratosthenes, formula.

How and why ...Statistics/统计学应用（英文版）   cnXiv2020100101P  [PDF]   167 pages      作者/Author： Charles Cheng Xia      发布于: 2020

摘要/Abstract： This book is written in Englis and focus on description of data and analysis of data with help you understand and learn exactly what you need to know about statistical ideas and techniques, fundamental formulas and calculations and statistical core topics in scope of applications. The book includes more than forty examples in explanation and/or illustration, step by step, to let you understand or have ideas to understand on how and why the statistic formula and calculation be applied.

本书采用英文撰写，聚焦于数据描述与数据分析。它旨在帮助读者精准理解并掌握核心统计学概念、方法、基础公式及计算，全面涵盖实际应用中的统计学核心课题。书中提供了四十多个循序渐进的案例解析与示范，帮助读者深刻领会统计公式与计算方法的应用场景与内在逻辑。

• Description of Data (数据描述 / 描述性统计): This involves summarizing and organizing data so it can be easily understood. It includes calculating things like the average (mean), the middle value (median), and how spread out the data is (standard deviation), or creating charts like histograms.
• Analysis of Data (数据分析 / 推断性统计): This goes beyond just describing what you have. It involves making predictions, finding trends, and drawing conclusions about a larger population based on a smaller sample of data (e.g., hypothesis testing or regression analysis).
• Fundamental Formulas and Calculations (基础公式与计算): These are the mathematical building blocks of statistics. Examples include the formulas for variance, probability, Z-scores, and confidence intervals.
• Scope of Applications (应用范围): Statistics is not just theoretical. This term means the book teaches you how to apply these math tools to real-world fields like business marketing, medical trials, social sciences, or quality control in manufacturing.

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