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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

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

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

High-dimensional robust inference for censored linear models

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Science China mathematics 2024年第4期67卷 891-918页

作者： Jiayu Huang Yuanshan Wu school of mathematics and statistics Wuhan UniversityWuhan 430072China school of statistics and mathematics Zhongnan University of Economics and LawWuhan 430073China

Due to the direct statistical interpretation,censored linear regression offers a valuable complement to the Cox proportional hazards regression in survival analysis.We propose a rank-based high-dimensional inference for censored linear regression without imposing any moment condition on the model error.We develop a theory of the high-dimensional U-statistic,circumvent challenges stemming from the non-smoothness of the loss function,and establish the convergence rate of the regularized estimator and the asymptotic normality of the resulting de-biased estimator as well as the consistency of the asymptotic variance estimation.As censoring can be viewed as a way of trimming,it strengthens the robustness of the rank-based high-dimensional inference,particularly for the heavy-tailed model error or the outlier in the presence of the response.We evaluate the finite-sample performance of the proposed method via extensive simulation studies and demonstrate its utility by applying it to a subcohort study from The Cancer Genome Atlas(TCGA).

关键词： censoring mechanism heavy-tailed distribution non-smooth loss function outlier rank regression

Semi-derived Ringel-Hall algebras and Hall algebras of odd-periodic relative derived categories

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Science China mathematics 2024年第8期67卷 1735-1760页

作者： Ji Lin Liangang Peng Department of mathematics and statistics Fuyang Normal University school of mathematics Sichuan University

Let t be a positive integer and A be a hereditary abelian category satisfying some finiteness conditions. We define the semi-derived Ringel-Hall algebra of A from the category CZ/t（A） of Z/t-graded complexes and obtain a natural basis of the semi-derived Ringel-Hall algebra. Moreover, we describe the semiderived Ringel-Hall algebra by the generators and defining relations. In particular, if t is an odd integer, we show an embedding of the derived Hall algebra of the odd-periodic relative derived category in the extended semi-derived Ringel-Hall algebra.

关键词： hereditary abelian categories the categories of $\mathbb{Z}/t$-graded complexes semi-derived Ringel-Hall algebras derived Hall algebras

On self-affine tiles that are homeomorphic to a ball

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Science China mathematics 2024年第1期67卷 45-76页

作者： Jorg M.Thuswaldner Shu-Qin Zhang mathematics and statistics University of LeobenLeoben 8700Austria school of mathematics and statistics Zhengzhou UniversityZhengzhou 450001China

Let M be a 3×3 integer matrix which is expanding in the sense that each of its eigenvalues is greater than 1 in modulus and let D?Z^(3)be a digit set containing|det M|elements.Then the unique nonempty compact set T=T(M,D)defined by the set equation MT=T+D is called an integral self-affine tile if its interior is nonempty.If D is of the form D={0,v,...,(|det M|-1)v},we say that T has a collinear digit set.The present paper is devoted to the topology of integral self-affine tiles with collinear digit sets.In particular,we prove that a large class of these tiles is homeomorphic to a closed 3-dimensional ball.Moreover,we show that in this case,T carries a natural CW complex structure that is defined in terms of the intersections of T with its neighbors in the lattice tiling{T+z:z∈Z^(3)}induced by T.This CW complex structure is isomorphic to the CW complex defined by the truncated octahedron.

关键词： self-afine sets tiles and tilings low-dimensional topology truncated octahedron