Subjects: Mathematics >> Control and Optimization. submitted time 2025-07-17
Abstract： 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.
Peer Review Status:
Awaiting Review
Subjects: Mathematics >> Discrete Mathematics and Combinatorics submitted time 2025-07-09
Abstract： In 1975, P. Erd\"{o}s proposed the problem of determining the
maximum number $f(n)$ of edges in a graph of $n$ vertices in
which any two cycles are of different
lengths. 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 $\sqrt
2$ (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
Subjects: Mathematics >> Numerical Analysis submitted time 2025-06-25
Abstract： 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.
Peer Review Status:Awaiting Review
Subjects: Information Science and Systems Science >> Basic Disciplines of Information Science and Systems Science Subjects: Mathematics >> Theoretical Computer Science Subjects: Physics >> Interdisciplinary Physics and Related Areas of Science and Technology submitted time 2025-06-11
Abstract： 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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Applied Mathematics submitted time 2025-06-09
Abstract： 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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Modeling and Simulation Subjects: Linguistics and Applied Linguistics >> Linguistics and Applied Linguistics Subjects: Computer Science >> Natural Language Understanding and Machine Translation submitted time 2025-06-03
Abstract： 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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics （General） submitted time 2025-05-30
Abstract： 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.
Subjects: Mathematics >> Computational Mathematics. submitted time 2025-05-30
Abstract： OpenPFEM is a parallel finite element computation platform developed in C, designed for large-scale parallel finite element computations on high-performance computers. This paper presents the fundamental framework of OpenPFEM, with a particular focus on the mesh distribution based parallel data communication method. On the sender side of communication, the data are organized and distributed according to the geometric distribution of the mesh, the data communication is unified based on the topological relationships of the sub-meshes. On the receiver side, the received data are processed according to the topological relationships of the sub-meshes, too. This paper demonstrates the application of this parallel data communication strategy through various examples, including computing the residual type of a posterior error estimator of finite element functions, calculation of stress-strain tensors of finite element functions, and reordering of interpolation matrices after the mesh load balancing.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Applied Mathematics submitted time 2025-05-19
Abstract： Recently, some window functions have been introduced into the nonuniform fast Fourier transform and the regularized Shannon sampling. Inspired by these works, we utilize a sinh-type function to accelerate the convergence of the Lagrangian nonuniform sampling series. Our theoretical error estimates and numerical experiments demonstrate that the sinh regularized nonuniform sampling series achieves a superior convergence rate compared to the fastest existing Gaussian regularized nonuniform sampling series.
Peer Review Status:Awaiting Review
Subjects: Statistics >> Economic Statistics Subjects: Mathematics >> Modeling and Simulation Subjects: Information Science and Systems Science >> Simulation Science and Technology submitted time 2025-05-17
Abstract： With China’s population aged 60 and above exceeding 290 million, smart old-age care, as an important part of the silver-haired economy, is faced with the realistic dilemma of "hot policy, cold market and idling technology".
In order to explore the impact of multi-agent collaboration mechanism on the evolution of smart elderly care system, this paper constructs a four-party evolutionary game model of government, enterprises, platforms and the elderly, and integrates system dynamics methods to explore the impact of multi-agent strategy game on the evolution of silver-haired economic system. Based on the benefit-cost function and replication dynamic equation, the model sets key variables such as government subsidy intensity, enterprise pricing strategy, platform data openness and acceptance willingness of the elderly, quantifies the behavior evolution mechanism, and uses Jacobian matrix to analyze strategy stability, identify strategy critical points and system stability conditions.
Based on Vensim platform, this paper designs six typical scenarios to simulate the system evolution process under different combinations of subsidies, pricing and open strategies. The simulation results show that the platform openness and the acceptance of the elderly constitute the key feedback path of the co-evolution of the system; Corporate behavior is affected by the linkage between platform incentives and user feedback, while the effect of government subsidies depends on the degree of strategic matching with market response. Under the scenario that the elderly don’t accept it, the system evolution shows strategy drift and non-equilibrium, which reflects the core role of game decision-making in the elderly group in the system stability.
Based on the above findings, this paper proposes a "three-chain collaboration mechanism" and a "strategic response index system" to provide quantitative basis and policy reference for the optimization of multi-subject strategies for smart elderly care and the high-quality development of the silver economy.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Control and Optimization. submitted time 2025-05-10
Abstract： 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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Control and Optimization. Subjects: Mathematics >> Algebra and Number Theory submitted time 2025-03-04
Abstract： 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
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematics （General） submitted time 2025-02-15
Abstract： By considering the spectrum of the Cesàro-Hardy operator on the Hilbert-Pólya space, we give a new way to understand the Riemann hypothesis for Riemann zeta function and Dirichlet L-function. We propose a key conjecture. The Riemann hypothesis is true under the conjecture.
Peer Review Status:Awaiting Review
Subjects: Physics >> General Physics: Statistical and Quantum Mechanics, Quantum Information, etc. Subjects: Mathematics >> Mathematical Physics Subjects: Computer Science >> Other Disciplines of Computer Science submitted time 2025-02-10
Abstract： Based on ’t Hooft’s principle of superposition of states beyond the usual one as described by Dirac in the conventional quantum mechanics, we present a topos-theoretic formalism of quantum artificial intelligence. At first, according to Turing’s test, we interpret an artificial intelligence (AI) system as a physical system described by a topos (which is a new physical theory builded by Isham {\it et al} in 2008). Secondly, by using measure theory, we construct a topos-theoretic model for a classical AI system such as deep learning; while we construct a topos-theoretic model for a quantum AI system by operator theory. Finally, we give the topos-theoretic description of a quantum neural network.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematical Physics submitted time 2024-12-28
Abstract： 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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Discrete Mathematics and Combinatorics submitted time 2024-12-23
Abstract： 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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Mathematical Physics submitted time 2024-12-19
Abstract： 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 the highest 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.
Peer Review Status:Awaiting Review
Subjects: Physics >> Geophysics, Astronomy, and Astrophysics Subjects: Mathematics >> Mathematical Physics submitted time 2024-12-10
Abstract： 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.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Algebra and Number Theory submitted time 2024-12-05
Abstract： The Stepwise Decomposition Method (SDM) refers to the process of decomposing a macroscopic entity into individual objects and describing the objects of the macroscopic entity through these individual objects. This idea is widely distributed in various aspects of life, technology, research, and more. This article takes the teaching of elementary transformations of matrices in Advanced Algebra as an example to discuss the stepwise decomposition method in mathematics teaching.
Peer Review Status:Awaiting Review
Subjects: Mathematics >> Theoretical Computer Science submitted time 2024-11-21
Abstract：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。
Peer Review Status:Awaiting Review
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