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Generic Existence of Infinitely Many Non-contractible Closed Geodesics on Compact Space Forms

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Acta Mathematica Sinica,English Series 2024年第7期40卷 1674-1684页

作者： Hui LIU Yu Chen WANG school of mathematics and statistics Wuhan University school of mathematics Tianjin Normal University

Let M = S~n/Γ and h be a nontrivial element of finite order p in π1（M）, where the integers n, p ≥ 2, Γ is a finite abelian group which acts freely and isometrically on the n-sphere and therefore M is diffeomorphic to a compact space form. In this paper, we prove that there are infinitely many noncontractible closed geodesics of class [h] on the compact space form with Cr-generic Finsler metrics,where 4 ≤ r ≤ ∞. The conclusion also holds for Cr-generic Riemannian metrics for 2 ≤ r ≤ ∞. The proof is based on the resonance identity of non-contractible closed geodesics on compact space forms.

关键词： Compact space forms non-contractible closed geodesics generic Finsler metrics resonance identity 53C22 58E05 58E10

A Mean-Field Game for a Forward-Backward Stochastic System With Partial Observation and Common Noise

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IEEE/CAA Journal of Automatica Sinica 2024年第3期11卷 746-759页

作者： Pengyan Huang Guangchen Wang Shujun Wang Hua Xiao the school of Control Science and Engineering Shandong University the school of statistics and mathematics Shandong University of Finance and Economics IEEE the school of Management Shandong University the school of mathematics and statistics Shandong University

This paper considers a linear-quadratic(LQ) meanfield game governed by a forward-backward stochastic system with partial observation and common noise,where a coupling structure enters state equations,cost functionals and observation equations.Firstly,to reduce the complexity of solving the meanfield game,a limiting control problem is introduced.By virtue of the decomposition approach,an admissible control set is proposed.Applying a filter technique and dimensional-expansion technique,a decentralized control strategy and a consistency condition system are derived,and the related solvability is also addressed.Secondly,we discuss an approximate Nash equilibrium property of the decentralized control strategy.Finally,we work out a financial problem with some numerical simulations.

关键词： Decentralized control strategy ϵ-Nash equilibrium forward-backward stochastic system mean-field game partial observation

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Adversarial data splitting for domain generalization

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Science China(Information Sciences) 2024年第5期67卷 28-42页

作者： Xiang GU Jian SUN Zongben XU school of mathematics and statistics Xi'an Jiaotong University

Domain generalization aims to learn a model that is generalizable to an unseen target domain,which is a fundamental and challenging task in machine learning for out-of-distribution generalization. This paper proposes a novel domain generalization approach that enforces the learned model to be able to generalize well over the train/val subset splitting of the training dataset. This idea is modeled herein as an adversarial data splitting framework, formulated as a min-max optimization problem inspired by the metalearning approach. The min-max optimization problem is solved by iteratively splitting the training dataset into the training and val subsets to maximize the domain shift measured by the objective function and updating the model parameters to enable the model to generalize well from the training subset to the val subset by minimizing the objective function. This adversarial training approach does not assume the known domain labels of the training data; instead, it automatically investigates the “hard” splitting of the train/val subsets to learn the generalizable model. Extensive experimental results using three benchmark datasets demonstrate the superiority of this approach. In addition, we derive a generalization error bound for the theoretical understanding of our proposed approach.

关键词： domain generalization adversarial learning data splitting meta-learning out of distribution

Zeros of Complete Symmetric Polynomials over Finite Fields

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Journal of Systems Science & Complexity 2024年第4期37卷 1772-1788页

作者： CAO Wei school of mathematics and statistics Minnan Normal University

Wan and Zhang(2021) obtained a nontrivial lower bound for the number of zeros of complete symmetric polynomials over finite fields, and proposed a problem whether their bound can be improved. In this paper, the author improves Wan-Zhang's bound from three aspects. The proposed results are based on the estimates related to the number of certain permutations and the value sets of non-permutation polynomials associated to the complete symmetric polynomial. And the author believes that there are still possibilities to improve the bounds and hence Wan-Zhang's bound.

关键词： Complete symmetric polynomial finite field permutation polynomial value set

Evolutionary game dynamics of combining two different aspiration-driven update rules in structured populations

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Chinese Physics B 2024年第5期33卷 182-191页

作者：杨智昊杨彦龙 mathematics and statistics school Guizhou UniversityGuiyang 550025China

In evolutionary games,most studies on finite populations have focused on a single updating mechanism.However,given the differences in individual cognition,individuals may change their strategies according to different updating mechanisms.For this reason,we consider two different aspiration-driven updating mechanisms in structured populations:satisfied-stay unsatisfied shift(SSUS)and satisfied-cooperate unsatisfied defect(SCUD).To simulate the game player’s learning process,this paper improves the particle swarm optimization algorithm,which will be used to simulate the game player’s strategy selection,i.e.,population particle swarm optimization(PPSO)algorithms.We find that in the prisoner’s dilemma,the conditions that SSUS facilitates the evolution of cooperation do not enable cooperation to emerge.In contrast,SCUD conditions that promote the evolution of cooperation enable cooperation to emerge.In addition,the invasion of SCUD individuals helps promote cooperation among SSUS individuals.Simulated by the PPSO algorithm,the theoretical approximation results are found to be consistent with the trend of change in the simulation results.

关键词： evolutionary game dynamics aspiration-driven update structured populations

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Self-powered and self-sensing devices based on piezoelectric energy harvesting

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Science China Technological Sciences 2024年第6期67卷 1631-1667页

作者： CHEN GanTong ZHU Yue HUANG DongMei ZHOU ShengXi school of Aeronautics Northwestern Polytechnical University school of mathematics and statistics Xidian University

Wearable devices, interactive human-machine interface equipment, wireless sensors, and small-scale cleaning devices play crucial roles in biomedical implantation, disease treatment, health monitoring, environmental purification, etc. These devices require a sustainable energy source to work effectively. With the consideration of the global energy crisis and environmental pollution, researchers are exploring new, stable, and environmentally friendly methods to power these low-powered devices.Mechanical energy is one of the most abundant natural energy sources. Converting mechanical energy from the ambient environment or host structures into electrical energy via the direct piezoelectric effect is an efficient energy harvesting technique.This paper reviews the application of advanced piezoelectric materials, and small-scale self-powered and self-sensing piezoelectric devices at the cubic centimeter scale in energy harvesting and health monitoring of human, animal, machinery, roads,bridges, as well as the pollutant degradation of the environment. Some of these devices have the capability to not only harvest mechanical energy but also enable real-time monitoring and analysis of the electrical signals generated by the direct piezoelectric effect, facilitating prompt decision-making and appropriate responses. In addition, potential challenges and future prospects of small-scale self-powered and self-sensing piezoelectric devices are discussed.

关键词： piezoelectric energy harvesting self-powered self-sensing wearable devices implantable devices

BANDED M-MATRIX SPLITTING PRECONDITIONER FOR RIESZ SPACE FRACTIONAL REACTION-DISPERSION EQUATION

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Journal of Computational mathematics 2024年第2期42卷 372-389页

作者： Shiping Tang Aili Yang Yujiang Wu school of mathematics and statistics Lanzhou UniversityLanzhou 730000China school of mathematics and statistics Hainan Normal UniversityHaikou 570000China

Based on the Crank-Nicolson and the weighted and shifted Grunwald operators,we present an implicit difference scheme for the Riesz space fractional reaction-dispersion equations and also analyze the stability and the convergence of this implicit difference scheme.However,after estimating the condition number of the coefficient matrix of the discretized scheme,we find that this coefficient matrix is ill-conditioned when the spatial mesh-size is sufficiently small.To overcome this deficiency,we further develop an effective banded M-matrix splitting preconditioner for the coefficient matrix.Some properties of this preconditioner together with its preconditioning effect are discussed.Finally,Numerical examples are employed to test the robustness and the effectiveness of the proposed preconditioner.

关键词： Riesz space fractional equations Toeplitz matrix conjugate gradient method Incomplete Cholesky decomposition Banded M-matrix splitting

THE SMOOTHING EFFECT IN SHARP GEVREY SPACE FOR THE SPATIALLY HOMOGENEOUS NON-CUTOFF BOLTZMANN EQUATIONS WITH A HARDPOTENTIAL

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Acta Mathematica Scientia 2024年第2期44卷 455-473页

作者：刘吕桥曾娟 school of mathematics and statistics Anhui Normal UniversityWuhu241002China school of mathematics and statistics Wuhan UniversityWuhan430072China

In this article, we study the smoothing effect of the Cauchy problem for the spatially homogeneous non-cutoff Boltzmann equation for hard potentials. It has long been suspected that the non-cutoff Boltzmann equation enjoys similar regularity properties as to whose of the fractional heat equation. We prove that any solution with mild regularity will become smooth in Gevrey class at positive time, with a sharp Gevrey index, depending on the angular singularity. Our proof relies on the elementary L^(2) weighted estimates.

关键词： Boltzmann equation Gevrey regularity non-cutoff hard potential

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The Complete Classification of Graphs Whose Second Largest Eigenvalue of the Eccentricity Matrix is Less than 1

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Acta Mathematica Sinica,English Series 2024年第7期40卷 1741-1766页

作者： Jian Feng WANG Xing Yu LEI Shu Chao LI Zoran STANI? school of mathematics and statistics Shandong University of Technology Faculty of mathematics and statistics Central China Normal University Faculty of mathematics University of Belgrade

The eccentricity matrix of a graph is obtained from the distance matrix by keeping the entries that are largest in their row or column, and replacing the remaining entries by zero. This matrix can be interpreted as an opposite to the adjacency matrix, which is on the contrary obtained from the distance matrix by keeping only the entries equal to 1. In the paper, we determine graphs having the second largest eigenvalue of eccentricity matrix less than 1.

关键词： Eccentricity matrix eigenvalue diameter principal submatrix forbidden subgraph 05C50