基于量子化数学的统一的Shannon信息度量理论

This project examines the Shannon entropy problems, such as information singularity and non-measurability of continuous entropy etc., and reveal the mathematical and physical reasons that cause these issues. Based on the original quantized mathematical theory, the unified uncertain discrete probability fields are defined and their relationship with the existing discrete and continuous probability fields are explained. Meanwhile, as a special case of physical models，the uncertain discrete probability distributions of the samplings for a stationary random radio signal formed by a set of photons are researched. Then, according to Shannon information axioms, we define the unified uncertain discrete information entropy function, deduce the modified form of Shannon continuous entropy in its average limit approximation, and reduced it to Shannon discrete entropy with the uncertain parameter degeneration. And, the lower limits of the average powers of signal and noise are deduced in Shannon formula. Thus, the problems of the information singularity and the divergence of ideal channel capacity are eliminated, and the unified theory of information measure is formed. Furthermore, we research the relationship between this information measure and other information measures based on uncertainty, and research the profound influence of information finiteness on physics. In addition, this project also supports part of follow-up works of the applicant’s study on gravity and cosmology based on quantized mathematical theory.

本项目研究Shannon信息熵定义的问题，即信息奇点以及连续熵的非度量性的问题，揭示造成这些问题的数学与物理的原因。基于独创的量子化数学定义统一的不确定离散概率场并说明其与现有的离散与连续的概率场的关系。同时，作为其物理模型的特例，研究由光子集合所形成的平稳随机无线电信号的抽样的不确定离散概率分布特性。接着，根据Shannon信息公理定义统一的不确定离散信息熵函数，由其平均极限近似可导出Shannon连续熵的修正形式，而通过其不确定参数的退化则可还原为Shannon离散熵。同时，由修正的连续熵函数导出Shannon公式中信号和噪声的平均功率的下限。消除信息奇点以及理想信道容量的发散性等问题，形成一个统一的信息度量理论。进而，研究该信息度量与其他基于不确定性的信息度量的关系，并研究信息有限性对于物理学的深刻影响。此外，本课题还支持申请者基于量子化数学理论的引力与宇宙学研究的部分后续工作。

本项目的研究进一步完善了作为引力量子化以及连续与离散的统一的概率与信息度量的前提和基础的量子化数学理论。在发展爱因斯坦广义相对论理论原理和理论体系，解决包括宇宙大爆炸和黑洞奇点、引力场和引力波能量、广义相对论与量子力学的兼容性、量子力学的波粒二象性、暗能量与暗物质以及真空灾难等重大物理学问题上取得了一系列重要的阶段性成果。在信息论在网络安全应用等方面的研究上取得了一系列有意义的研究结果。所提出的恶意代码可视化检测和分类方法发展了应对网络攻击的信息安全技术，并从视觉表现、识别准确率和识别效率等方面上大大提升了应对恶意代码攻击的有效性。