分类： 物理学 >> 普通物理:统计和量子力学,量子信息等 分类： 数学 >> 数学物理 分类： 计算机科学 >> 计算机科学技术其他学科 提交时间： 2025-02-10
摘要： 依据’t Hooft在2016年提出的关于量子态叠加原理的新诠释, 本文运用拓扑斯理论研究量子人工智能的数学基础. 首先, 根据图灵测试对智能给出的科学定义, 我们将人工智能系统定义为由拓扑斯理论所描述的物理系统(Isham等人在2008年建立的物理理论), 它们具有自身的高阶形式语言及逻辑推理系统. 其次, 依据物理原理, 我们将人工智能系统分为经典人工智能系统和量子人工智能系统. 对于经典人工智能系统, 我们用测度论构造相应的拓扑斯描述, 特别是给出了深度学习的拓扑斯描述; 而对于量子人工智能系统, 则用Hilbert空间上算子理论构造相应的拓扑斯以给出它的数学描述. 最后, 本文给出了量子神经网络的拓扑斯理论模型.
分类： 数学 >> 数学物理 提交时间： 2024-12-28
摘要： Objective:This article aims to construct a general formula from a philosophical perspective to describe the connections between the common attributes of things, providing preliminary insights into complex phenomena such as chaos theory and deep learning.Methods: Through conceptual analysis and theoretical deduction, a mathematical model based on attribute quantities is established to quantify the connections between things.Results:The study finds that the formula is applicable to analyzing chaotic phenomena in physics and complex systems in social sciences, potentially aiding in explaining difficulties in human learning and the accuracy improvement of deep learning models.Limitations:The theoretical construction lacks extensive empirical support, and the determination of attribute quantities and standards poses practical challenges.Conclusions:The common attribute connection formula offers a new tool for understanding the connections between things, and future research should focus on empirical validation and interdisciplinary applications.
分类： 数学 >> 数学物理 提交时间： 2024-12-19
摘要： In this paper, the Hopf cyclicity and hidden chaos for the three-dimensional (3D) $Z_2$-symmetric R\"{o}ssler system are investigated. Applying the recursive formula of the singular point quantities, and by strict symbolic calculation, we determine thehighest order three of weak focus at the symmetric equilibria on center manifold. And under suitable perturbation, six and at most six small amplitude limit cycles can generate from symmetric equilibria via Hopf bifurcation. Furthermore, we study different cases that multiple Hopf bifurcation and chaos can simultaneously occur around the two symmetric equilibria, where one (2, 2) distribution of four limit cycles is accompanied by chaos. To our knowledge, this property is very rare in many chaotic systems.
分类： 物理学 >> 地球物理学、天文学和天体物理学 分类： 数学 >> 数学物理 提交时间： 2024-12-10
摘要： Photon sphere has attracted considerable interests in the studies of black hole and other astrophysical objects. For different categories of spacetimes (or gravitational sources), the existence of photon spheres and their distributions are dramatically influenced by geometric and topological properties of spacetimes and characteristics of the corresponding gravitational fields. In this work, we carry out a geometric analysis on photon spheres for different categories of spacetime (including black hole spacetime, ultra-compact object’s spacetime, regular spacetime, naked singularity spacetime). Some universal properties and conclusions are obtained for these spacetimes. We mostly focus on the existence of photon spheres, total number of photon spheres $n = n_{\text{stable}} + n_{\text{unstable}}$, the subtraction of stable photon sphere and unstable photon sphere $w = n_{\text{stable}} - n_{\text{unstable}}$ in different categories of spacetimes. These conclusions are derived solely from geometric properties of optical geometry of spacetimes, irrelevant to the specific spacetime metric forms. Besides, our results successfully recover some important theorems on photon spheres proposed in recent years.
分类： 数学 >> 数学物理 提交时间： 2022-12-07
摘要： In this article we employ classical tricks to give local and global well-posedness to MagnetoElasticity System. Different from many cases, we consider the equation which the magnetic field satisfies is Landau-Lifshitz system without viscidity, i.e. the Schrodinger flow. As is well known, people can not obtain global existence of Schrodinger flow at general cases. However, the reason why we do what others can not do is the Schrodinger flow with non-zero convection term.
分类： 数学 >> 数学物理 提交时间： 2021-06-16
摘要： In this paper, we mainly discuss the copositivity of 4th order symmetric tensor defined by scalar dark matter stable under a $\mathbb{Z}_{3}$ discrete group, and obtain an analytically necessary and sufficient condition of the copositivity of such a class of tensors. Furthermore, this analytic expression may be used to verify the vacuum stability for $\mathbb{Z}_{3}$ scalar dark matter.
分类： 数学 >> 数学物理 提交时间： 2019-11-23
摘要： The strict opositivity of 4th order symmetric tensor may apply to detect vacuum stability of general scalar potential. For finding analytical expressions of (strict) opositivity of 4th order symmetric tensor, we may reduce its order to 3rd order to better deal with it. So, it is provided that several analytically sufficient conditions for the copositivity of 3th order 2 dimensional (3 dimensional) symmetric tensors. Subsequently, applying these conclusions to 4th order tensors, the analytically sufficient conditions of copositivity are proved for 4th order 2 dimensional and 3 dimensional symmetric tensors. Finally, we apply these results to present analytical vacuum stability conditions for vacuum stability for $\mathbb{Z}_3$ scalar dark matter.
分类： 物理学 >> 基本粒子与场物理学 分类： 数学 >> 数学物理 提交时间： 2018-10-08
摘要： 本文系统地探讨了霍奇星算子与外微分算符作用于任意微分形式场时两者的一般组合规律。首先，找到了保持微分形式场的次不变的两个组合算符，并通过二者的线性组合得到了一个新算符。其次，当由任意数目的霍奇星算子与外微分算符进行组合时，作者导出了所有形式上彼此互异的组合算符的统一表达式，这些表达式由单个霍奇星算子与外微分算符以及二者的任选两个的非零组合构成。在此基础上，分析了所有算符之间的相互作用关系，并根据这些算符对微分形式的次的改变情况，对它们进行了具体分类。最后，作为一个应用，作者详细讨论了如何由次相同的微分形式的线性组合来构造电磁场的麦克斯韦方程。
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