分类： 数学 >> 应用数学 提交时间： 2025-05-19
摘要： 最近，一些窗函数被引入到非均匀快速傅里叶变换和正则化香农采样中。受这些工作的启发，我们利用sinh型函数来加速拉格朗日非均匀采样序列的收敛。我们的理论误差估计和数值实验表明，与现有最快的高斯正则化非均匀采样序列相比，sinh正则化非一致采样序列的收敛速度更快。
分类： 统计学 >> 经济统计学 分类： 数学 >> 建模与仿真 分类： 信息科学与系统科学 >> 仿真科学技术 提交时间： 2025-05-17
摘要： 随着我国60岁及以上人口突破2.9亿，智慧养老作为银发经济的重要组成部分，面临”政策热、市场冷、技术空转”的现实困境。为探究多主体协同机制对智慧养老系统演化的影响，本文构建了政府、企业、平台与老年人四方演化博弈模型，融合系统动力学方法，探讨多主体策略博弈对银发经济系统演化的影响。模型以收益-成本函数与复制动态方程为基础，设定政府补贴强度、企业定价策略、平台数据开放度与老年人接纳意愿等关键变量，量化行为演化机制，并利用Jacobian矩阵分析策略稳定性，识别策略临界点与系统稳定性条件。基于Vensim平台，本文设计六类典型情景，模拟不同补贴、定价与开放策略组合下的系统演化过程。仿真结果表明，平台开放与老年人接纳构成系统协同演化的关键反馈路径；企业行为受平台激励和用户反馈联动影响，政府补贴效果则依赖于与市场响应之间的策略匹配度。在老年人不接纳情景下，系统演化出现策略漂移与非均衡态，反映出老年群体博弈决策在系统稳定性中的核心作用。基于上述发现，本文提出“三链协同机制”与“策略响应指标体系”，为智慧养老多元主体策略优化与银发经济高质量发展提供定量依据与政策参考。
分类： 数学 >> 控制和优化 提交时间： 2025-05-10
摘要： For the ideal transmission line governed by telegrapher’s equations, a mixed finite element method, a generalization of popular spatially discretized schemes, has been proposed. This numerical approximation scheme preserves both the Dirac structure and passivity, ensuring that the spatially discretized system retains its port-Hamiltonian nature. In this paper, we apply this method to spatially discretize two infinite-dimensional port-Hamiltonian systems with variable coefficients and boundary controls. We then investigate the preservation of exponential stability in the resulting semi-discretized systems, demonstrating their uniform exponential stability with respect to discretization parameters. For both semi-discretized models, the uniform exponential stabilities are derived through frequency domain analysis. Finally, numerical simulations validate the effectiveness of this semi-discrete scheme.
分类： 数学 >> 控制和优化 分类： 数学 >> 代数与数论 提交时间： 2025-03-04
摘要： It is well-known that a symmetric matrix with its entries \pm1 is not positive definite. But this is not ture for symmetric tensors (hyper-matrix). In this paper, we mainly dicuss the positive (semi-)definiteness criterion of a class of 4th order 3-dimensional symmetric tensors with entries t_{ijkl}\in\{-1,0,1\}. Through theoretical derivations and detailed classification discussions, the criterion for determining the positive (semi-)definiteness of such a class of tensors are provided based on the relationships and number values of its entries. Which establishes some unique properties of higher symmetric tensors that distinct from ones of matrces
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 >> 数值分析 提交时间： 2025-01-08
摘要： We introduce a novel random integration algorithm that boasts both high convergence order and polynomial tractability for functions characterized by sparse frequencies or rapidly decaying Fourier coefficients. Specifically, for integration in periodic isotropic Sobolev space and the isotropic Sobolev space with compact support, our approach attains a near-optimal root mean square error. In contrast to previous nearly optimal algorithms, our method exhibits polynomial tractability,ensuring that the number of samples does not scale exponentially with increasing dimensions. Our integration algorithm also enjoys near-optimal bound for weighted Korobov space. Furthermore, the algorithm can be applied without the need for prior knowledge of weights, distinguishing it from component-by-component algorithms. For integration in the Wiener algebra, the sample complexity of our algorithm is independent of the decay rate of Fourier coefficients. The effectiveness of the integration is confirmed through numerical experiments.
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 >> 离散数学和组合数学 提交时间： 2024-12-23
摘要： Recently, two extraordinary results on aperiodic monotiles have been obtained in two different settings. One is a family of aperiodic monotiles in the plane discovered by Smith, Myers, Kaplan and Goodman-Strauss in 2023, where rotation is allowed, breaking the 50-year-old record (aperiodic sets of two tiles found by Roger Penrose in the 1970s) on the minimum size of aperiodic sets in the plane. The other is the existence of an aperiodic monotile in the translational tiling of \mathbb{Z}^n for some huge dimension n proved by Greenfeld and Tao. This disproves the long-standing periodic tiling conjecture. However, it is known that there is no aperiodic monotile for translational tiling of the plane. The smallest size of known aperiodic sets for translational tilings of the plane is 8, which was discovered more than 30 years ago by Ammann. In this paper, we prove that translational tiling of the plane with a set of 7 polyominoes is undecidable. As a consequence of the undecidability, we have constructed a family of aperiodic sets of size 7 for the translational tiling of the plane. This breaks the 30-year-old record of Ammann.
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 >> 代数与数论 提交时间： 2024-12-05
摘要： 逐步分解法（SDM=Stepwise Decomposition Method）指的是通过将一个宏观整体的 对象分解为一些单一对象，并通过这些单一对象来描述宏观整体的对象。这一思想广泛分 布在生活、科技、研究等方方面面。本文以《高等代数》中矩阵的初等变换的教学为例， 对数学教学中的逐步分解法展开一些论述。
分类： 数学 >> 控制和优化 提交时间： 2024-11-12
摘要： This paper focuses on the strict copositivity analysis of 4th-order 3-dimensional symmetric tensors. A necessary and sufficient condition is provided for the strict copositivity of a fourth-order symmetric tensor. Subsequently, building upon this conclusion, we discuss the strict copositivity of fourth-order three-dimensional symmetric tensors with its entries \pm 1, 0, and further build their necessary and sufficient conditions. Utilizing these theorems, we can effectively verify the strict copositivity of a general fourth-order three-dimensional symmetric tensors.
分类： 数学 >> 代数与数论 分类： 数学 >> 离散数学和组合数学 提交时间： 2024-11-10
摘要： 本文建立了有限么半群的表示范畴与有限维代数的模范畴之间的联系．进一步地，利用此联系证明了存在有限维基代数，使得定义在此代数上的交替弦模的同构类数有限，此代数的箭图除源点和沉点以外的其它顶点均为结点，且该代数表示无限，
分类： 数学 >> 数学（综合） 提交时间： 2024-11-01
摘要： 目的：探讨A,B,P是圆锥曲线C上不同的三点，探讨直线PA,PB的斜率的积（和）为定值与直线AB是否经过定点的关系.方法：设而不求，恰当转化，通过斜率齐次化简化计算，探索发现.结论：在特定条件下，直线PA,PB的斜率的和（积）为定值等价于直线AB经过定点,据此可以创编若干具体问题.
分类： 数学 >> 代数与数论 提交时间： 2024-10-18
摘要： 本文主要研究如何教好高等代数课程。通过对湖北大学数学与统计学学院高等代数教学的实践总结和反思，为了更好地实现课堂的教学，教师应该注意的问题包括吃透教材，注意大学生的年龄段特点，采取恰当的教学方法，注意学生的实际情况，充分了解学生，讲课和讲题要相互配合，因材施教，注意学生兴趣的培养，启发式教学等。关注这些问题将有助于教学质量的提高。
分类： 数学 >> 代数与数论 提交时间： 2024-10-18
摘要： 在数学证明中，数学归纳法是一种常见、重要的证明方法。这篇论文归纳总结了高等代数证明中适用数学归纳法的类型。这些类型包括：直接对正整数作归纳，对多项式的次数作归纳，对行列式的阶数作归纳，对方阵的阶数作归纳，对向量空间的维数作归纳。
分类： 数学 >> 代数与数论 提交时间： 2024-10-18
摘要： 高等代数是高等院校数学专业的一门重要的基础课程。通过这一课程的教学，使学生系统地掌握一元多项式和线性代数基础理论和基本方法，提高学生数学抽象、逻辑推理、发现问题、解决问题的能力。本文以欧式除法为例介绍高等代数课堂教学设计。
分类： 数学 >> 代数与数论 分类： 数学 >> 离散数学和组合数学 提交时间： 2024-10-15
摘要： 令An是 2 次Jacobson根为零的A型Nakayama代数, Xn 是 2 次Jacobson根为零的 tilde{A}型Nakayama代数. 本文考虑了An与Xn的k-张量An tensor Xn上的不可分解模的分类问题, 并给出其在同构意义下的计数公式.
分类： 数学 >> 控制和优化 提交时间： 2024-10-12
摘要： In this article, we mainly give the strictly copositive conditions of a special class of third order three dimensional symmetric tensors. More specifically, by means of the polynomial decomposition method, the analytic sufficient and necessary conditions are established for checking the strict copositivity of a 3rd order 3-dimensional symmetric tensor with its entries in \{-1,0,1\}. Several strict inequalities of cubic ternary homogeneous polynomials are presented by applying these conclusions. Some criteria which ensure the strict copositivity of a general 3rd order 3-dimensional tensor are obtained
分类： 数学 >> 控制和优化 分类： 信息科学与系统科学 >> 信息科学与系统科学基础学科 分类： 数学 >> 建模与仿真 提交时间： 2024-10-09
摘要： Modern mathematical neural networks are derived from biological neural networks, yet the currently popular general large models do not incorporate biological neural networks. The primary reason for this is that the differential equations based on biological neural networks are difficult to manipulate. At present, mathematical neural networks are characterized by their capacity for large-scale deployment, while biological neural networks offer strong biological interpretability. This paper introduces a system of differential equations with perfect symmetry and convenient manipulability, enabling us to manipulate this system as easily as we manipulate numbers in a matrix, thus integrating the advantages of both. As we are introducing a brand-new neural network framework, we first explore the mathematical properties of the differential equations, then define a new signal propagation method, and finally propose a new training approach for the neural network. The training of this new neural network does not rely on the traditional back-propagation algorithm; instead, it depends solely on the propagation of local signals. This implies that we no longer require global information to train the network. Each neuron can adjust based on the signals it receives and its predetermined strategy. As a verification, we mimicked the linking method of a multilayer perceptron (MLP) to create a new neural network and trained it on the MNIST dataset, demonstrating the effectiveness of our methodology.
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