分类： 数学 >> 控制和优化 提交时间： 2025-07-17
摘要： Based on the tensor-based large margin distribution and the nonparallel support tensor machine, we establish a novel classifier for binary classification problem in this paper, termed the Large Margin Distribution based NonParallel Support Tensor Machine (LDM-NPSTM). The proposed classifier has the following advantages: First, it utilizes tensor data as training samples, which helps to comprehensively preserve the inherent structural information of high-dimensional data, thereby improving classification accuracy. Second, this classifier not only considers traditional empirical risk and structural risk but also incorporates the marginal distribution information of the samples, further enhancing its classification performance. To solve this classifier, we use alternative projection algorithm. Specifically, building on the formulation where in the proposed LDM-NPSTM, the parameters defining the separating hyperplane form a tensor (tensorplane) constrained to be the sum of rank-one tensors, the corresponding optimization problem is solved iteratively using alternative projection algorithm. In each iteration, the parameters related to the projections along a single tensor mode are estimated by solving a typical Support Vector Machine-type optimization problem. Finally, the efficiency and performance of the proposed model and algorithm are verified through theoretical analysis and some numerical examples.
分类： 数学 >> 离散数学和组合数学 提交时间： 2025-07-09
摘要： In 1975, P. Erd\"{o}s proposed the problem of determining themaximum number $f(n)$ of edges in a graph of $n$ vertices inwhich any two cycles are of differentlengths. In this paper, it is proved that $$f(n)\geq n+36t$$ for $t=1260r+169 \,\ (r\geq 1)$and $n \geq 540t^{2}+\frac{175811}{2}t+\frac{7989}{2}$. Consequently,$\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 +{2 \over 5}},$ which is better than the previous bounds $\sqrt2$ (see [2]), $\sqrt {2+{2562\over 6911}}$ (see [7]). \par Combining this with Boros, Caro, F\"uredi and Yuster’s upper bound, we get $$1.98\geq \limsup_{n \rightarrow \infty} {f(n)-n\over \sqrt n} \geq \liminf_{n \rightarrow \infty} {f(n)-n\over \sqrt n}\geq \sqrt {2.4}.$$ \par
分类： 数学 >> 数值分析 提交时间： 2025-06-25
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： 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 >> 计算科学理论 分类： 物理学 >> 交叉学科物理及相关领域的科学与技术 提交时间： 2025-06-11
摘要： We present a new method for structural sequence analysis grounded in Algorithmic Information Theory (AIT). At its core is the Ladderpath approach, which extracts nested and hierarchical relationships among repeated substructures in linguistic sequences---an instantiation of AIT’s principle of describing data through minimal generative programs. These structures are then used to define three distance measures: a normalized compression distance (NCD), and two alternative distances derived directly from the Ladderpath representation. Integrated with a $k$-nearest neighbor classifier, these distances achieve strong and consistent performance across in-distribution, out-of-distribution (OOD), and few-shot text classification tasks. In particular, all three methods outperform both gzip-based NCD and BERT under OOD and low-resource settings. These results demonstrate that the structured representations captured by Ladderpath preserve intrinsic properties of sequences and provide a lightweight, interpretable, and training-free alternative for text modeling. This work highlights the potential of AIT-based approaches for structural and domain-agnostic sequence understanding.
分类： 数学 >> 应用数学 提交时间： 2025-06-09
摘要： A fractional-order proportional-derivative controller is designed to address bifurcation issues in a dual-time-delay fractional-order predator-prey model. By selecting different delays as bifurcation parameters, the stability and Hopf bifurcation conditions of the controlled system are derived. The results show that the fractional order, delays and control parameters play an important role on the stability and Hopf bifurcation of the system. By selecting reasonable system parameters (fractional order, delays, and control parameters), suitable system control strategies can be devised. Finally, the key findings of this study are verified through numerical examples.
分类： 数学 >> 建模与仿真 分类： 语言学及应用语言学 >> 语言学及应用语言学 分类： 计算机科学 >> 自然语言理解与机器翻译 提交时间： 2025-06-03
摘要： Natural language is considered closely intertwined with human cognition, with linguistic structures posited to offer profound insights into the cognitive system. However, as a coding system, natural language encodes diverse objects into unified forms; its prominent formal features capture people’s attention, such as lexical combinatorial rules, which tend to overshadow those form-independent structures. Here, I present knowledge-level, logic-level, task-level, and model-level semantic structures inherent in natural language. These structures are discovered by shifting the research focus from coding forms of natural language to the objects they encode, unveiling different semantic layers integrated within sentences. The cognitive functions of these structures are evident both in themselves and in models developed from them. I therefore introduce four models to demonstrate their capabilities in memorization, reasoning, learning, natural language generation, and understanding. These findings advance our understanding of natural language and provide a framework for investigating the cognitive system’s information processing through structural analysis of natural language.
分类： 数学 >> 数学（综合） 提交时间： 2025-05-30
摘要： Keyboard-based interaction may not accommodate various needs, especially for individuals with disabilities. While inertial sensor-based writing recognition is promising due to the sensors’ small size, wearability, and low cost, accurate recognition in the Chinese context is hampered by the difficulty of collecting extensive inertial signal samples for the vast number of characters. Therefore, we design a Chinese Inertial GAN (CI-GAN) containing Chinese glyph encoding (CGE), forced optimal transport (FOT), and semantic relevance alignment (SRA) to acquire unlimited high-quality training samples. Unlike existing vectorization methods focusing on the meaning of Chinese characters, CGE represents shape and stroke features, providing glyph guidance for writing signal generation. FOT establishes a triple-consistency constraint between the input prompt, output signal features, and real signal features, ensuring the authenticity and semantic accuracy of the generated signals. SRA aligns semantic relationships between multiple outputs and their input prompts, ensuring that similar inputs correspond to similar outputs (and vice versa), alleviating model hallucination. The three modules guide the generator while also interacting with each other, forming a coupled system. By utilizing the massive training samples provided by CI-GAN, the performance of six widely used classifiers is improved from 6.7% to 98.4%, indicating that CI-GAN constructs a flexible and efficient data platform for Chinese inertial writing recognition. Furthermore, we release the first Chinese inertial writing dataset on GitHub.
分类： 数学 >> 计算数学 提交时间： 2025-05-30
摘要： OpenPFEM 是一款以C语言开发的并行有限元计算平台，旨在高性能计算机上进行大规模的并行有限元计算.本文介绍了OpenPFEM的基本建构，尤其重点介绍了我们所设计的基于网格划分的并行数据通信方式．在数据传输的发送端将需要通信的数据按照网格的几何元素进行组织和分布，并根据各进程子网格的拓扑关系统一进行数据传输：在数据接收端，同样根据各进程子网格的拓扑关系对接收到的数据进行处理.本文通过计算有限元函数的残量型后验误差估计、有限元函数的应力应变张量和网格负载均衡后插值矩阵的重排等例子展示了这一并行数据传输策略的应用.
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 $4$th 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： 数学 >> 数学（综合） 提交时间： 2025-02-15
摘要： By considering the he spectrum of the Cesaro operator on the Hilber-Polya space, we proved the Riemann hypothesis for Riemann zeta function and Dirichlet L-function.
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 >> 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 >> 计算科学理论 提交时间： 2024-11-21
摘要： 虚拟数字人作为数字经济新的增长点，其具有“虚拟+现实”的特征，以多元化的姿态渗透在影视传媒、直播营销、金融投资等各种行业中，与现实社会产生交互。金融行业作为数字人产业发展的重要领域，其具有良好的发展前景，各金融机构纷纷加快金融战略部署，布局“数字人”金融场景。但是在虚拟数字人快速应用到金融行业的同时，也伴随着算法偏见、数据合规、侵权责任、市场准入的法律风险。本文深入地分析了数字人金融应用过程中存在的法律关系，以现存的法律风险为治理模块，审视上述风险下开发者、金融机构、投资者、监管者的治理义务，通过利益相关者理论构建虚拟数字人金融应用的主体责任治理机制，厘清了各种法律风险下多元主体的治理责任。
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