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Rational Solutions of First Order Algebraic Ordinary Differential Equations

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Journal of Systems Science & Complexity 2024年第2期37卷 567-580页

作者： FENG Shuang SHEN Liyong school of Physical and mathematical sciences Nanjing Tech UniversityNanjing 211816China school of mathematical sciences University of Chinese Academy of SciencesBeijing 100049China

Let f(t,y,y')=∑ _(i=0)^(n )a_(i)(t,y)y'^(i)=0 be an irreducible first order ordinary differential equation with polynomial coefficients.Eremenko in 1998 proved that there exists a constant C such that every rational solution of f(t,y,y')=0 is of degree not greater than C.Examples show that this degree bound C depends not only on the degrees of f in t,y,y' but also on the coefficients of f viewed as the polynomial in t,y,y'.In this paper,the authors show that if f satisfies deg(f,y)0,then the degree bound C only depends on the degrees of f in t,y,y',and furthermore we present an explicit expression for C in terms of the degrees of f in t,y,y'.

关键词： Degree bound first order AODE height rational solution

Geometry of holomorphic invariant strongly pseudoconvex complex Finsler metrics on the classical domains

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Science China Mathematics 2024年第8期67卷 1827-1864页

作者： Xiaoshu Ge Chunping Zhong school of mathematical sciences Xiamen University

In this paper, a class of holomorphic invariant metrics is introduced on the irreducible classical domains of types Ⅰ-Ⅳ, which are strongly pseudoconvex complex Finsler metrics in the strict sense of Abate and Patrizio(1994). These metrics are of particular interest in several complex variables since they are holomorphic invariant complex Finsler metrics found in the literature which enjoy good regularity as well as strong pseudoconvexity and can be explicitly expressed to admit differential geometry studies. They are,however, not necessarily Hermitian quadratic as the Bergman metrics. These metrics are explicitly constructed via deformation of the corresponding Bergman metric on the irreducible classical domains of types Ⅰ-Ⅳ,respectively, and they are all proved to be complete K?hler-Berwald metrics. They enjoy very similar curvature properties as those of the Bergman metric on the irreducible classical domains, i.e., their holomorphic sectional curvatures are bounded between two negative constants and their holomorphic bisectional curvatures are always nonpositive and bounded below by negative constants, respectively. From the viewpoint of complex analysis,these metrics are analogs of Bergman metrics in complex Finsler geometry which do not necessarily have Hermitian quadratic restrictions in the sense of Chern(1996).

关键词： holomorphic invariant metric Kähler-Berwald metric irreducible classical domains

Data-Driven Direct Adaptive Risk-Sensitive Control of Stochastic Systems

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Journal of Systems Science & Complexity 2024年第4期37卷 1446-1469页

作者： QIAO Nan LI Tao school of mathematical sciences East China Normal University Shanghai Key Laboratory of Pure Mathematics and mathematical Practice School of Mathematical SciencesEast China Normal University

The authors propose a data-driven direct adaptive control law based on the adaptive dynamic programming(ADP) algorithm for continuous-time stochastic linear systems with partially unknown system dynamics and infinite horizon quadratic risk-sensitive indices. The authors use online data of the system to iteratively solve the generalized algebraic Riccati equation(GARE) and to learn the optimal control law directly. For the case with measurable system noises, the authors show that the adaptive control law approximates the optimal control law as time goes on. For the case with unmeasurable system noises, the authors use the least-square solution calculated only from the measurable data instead of the real solution of the regression equation to iteratively solve the GARE. The authors also study the influences of the intensity of the system noises, the intensity of the exploration noises, the initial iterative matrix, and the sampling period on the convergence of the ADP algorithm.Finally, the authors present two numerical simulation examples to demonstrate the effectiveness of the proposed algorithms.

关键词： Adaptive dynamic programming direct adaptive control generalized algebraic Riccati equation risk-sensitive control

Global well-posedness for 2D inhomogeneous asymmetric fluids with large initial data

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Science China Mathematics 2024年第3期67卷 527-556页

作者： Chenyin Qian Beibei He Ting Zhang school of mathematical sciences Zhejiang Normal UniversityJinhua 321004China school of mathematical sciences Zhejiang UniversityHangzhou 310058China

In this paper, by using time-weighted global estimates and the Lagrangian approach, we first investigate the global existence and uniqueness of the solution for the 2D inhomogeneous incompressible asymmetric fluids wi... 详细信息

关键词： inhomogeneous asymmetric fluids Littlewood-Paley theory Besov spaces global well-posedness

Partial regularity and nonlinear potential estimates for Stokes systems with super-quadratic growth

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Science China Mathematics 2024年第7期67卷 1525-1554页

作者： Lingwei Ma Zhenqiu Zhang school of mathematical sciences Tianjin Normal UniversityTianjin300387China school of mathematical sciences and LPMC Nankai UniversityTianjin300071China

This paper builds a bridge between partial regularity theory and nonlinear potential theory for the following generalized stationary Stokes system with super-quadratic growth and continuous coefficients:-div A(x,Du)+■π=f,div u=0,where Du is the symmetric part of the gradient■u.We first establish anε-regularity criterion involving both the excess functional of the symmetric gradient Du and Wolff potentials of the nonhomogeneous term f to guarantee the local vanishing mean oscillation(VMO)-regularity of Du in an open subset Ω_(u) of Ω with full measure.Such anε-regularity criterion leads to a pointwise Wolff potential estimate of Du,which immediately infers that Du is partially C^(0)-regular under appropriate assumptions.Finally,we give a local continuous modulus estimate of Du.

关键词： nonlinear Stokes system partial regularity Wolff potential estimates continuous modulus estimates

NeuroPrim:An attention-based model for solving NP-hard spanning tree problems

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Science China Mathematics 2024年第6期67卷 1359-1376页

作者： Yuchen Shi Congying Han Tiande Guo school of mathematical sciences University of Chinese Academy of SciencesBeijing100049China

Spanning tree problems with specialized constraints can be difficult to solve in real-world scenarios,often requiring intricate algorithmic design and exponential time.Recently,there has been growing interest in end-to-end deep neural networks for solving routing problems.However,such methods typically produce sequences of vertices,which make it difficult to apply them to general combinatorial optimization problems where the solution set consists of edges,as in various spanning tree problems.In this paper,we propose NeuroPrim,a novel framework for solving various spanning tree problems by defining a Markov decision process for general combinatorial optimization problems on graphs.Our approach reduces the action and state space using Prim's algorithm and trains the resulting model using REINFORCE.We apply our framework to three difficult problems on the Euclidean space:the degree-constrained minimum spanning tree problem,the minimum routing cost spanning tree problem and the Steiner tree problem in graphs.Experimental results on literature instances demonstrate that our model outperforms strong heuristics and achieves small optimality gaps of up to 250 vertices.Additionally,we find that our model has strong generalization ability with no significant degradation observed on problem instances as large as 1,000.Our results suggest that our framework can be effective for solving a wide range of combinatorial optimization problems beyond spanning tree problems.

关键词： degree-constrained minimum spanning tree problem minimum routing cost spanning tree problem Steiner tree problem in graphs Prim's algorithm reinforcement learning

On a Flexible Loan Repayment Method Depending on Borrower's Asset with an Early Termination Clause

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Acta Mathematicae Applicatae Sinica 2024年第2期40卷 557-567页

作者： Jin LIANG Jia-qi MAO Zhao-ya LIU school of mathematical sciences Tongji UniversityShanghai 200092China

Aiming at the problem that the asset’s fluctuation influences the borrower’s repayment ability,a loan with a new and flexible repayment method is designed,which depends on the asset value of the borrower.The repayment method can reduce the loan default probability,but it causes the uncertainty of the pay off time.Because the repayment term is related to the regular repayment amount in this method,a boundary for the regular repayment amount is set up in order to avoid too long repayment term.This will balance the benefit of borrowers and lenders and improve the applicability of this method.By establishing a mathematical model of the residual value of the loan,this model can be transformed into an initial-boundary problem of a partial differential equation.The analytic solution and the expected time to pay off the loan are obtained.Finally,numerical analysis are shown.

关键词： flexible repayment method repayment ratio early termination clause expected pay off time

Shock Formation for 2D Isentropic Euler Equations with Self-similar Variables

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Chinese Annals of Mathematics,Series B 2024年第3期45卷 349-412页

作者： Wenze SU school of mathematical sciences Fudan UniversityShanghai 200433China

The author studies the 2D isentropic Euler equations with the ideal gas law.He exhibits a set of smooth initial data that give rise to shock formation at a single point near the planar symmetry.These solutions to the 2D isentropic Euler equations are associated with non-zero vorticity at the shock and have uniform-in-time-1/3-Holder bound.Moreover,these point shocks are of self-similar type and share the same profile,which is a solution to the 2D self-similar Burgers equation.The proof of the solutions,following the 3D construction of Buckmaster,Shkoller and Vicol(in 2023),is based on the stable 2D self-similar Burgers profile and the modulation method.

关键词： 2D isentropic Euler equations Shock formation Self-similar solution

The Strategic Analysis of Demand Forecast-Sharing in a Hybrid-Format Online Platform Supply Chain

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Journal of Systems Science and Systems Engineering 2024年第3期33卷 281-310页

作者： Jing Zhao Zijun Yin Guobiao Zhou school of mathematical sciences Tiangong UniversityTianjin300387China

This paper examines the optimal forecast-sharing strategy in a hybrid-format online platform supply chain where a supplier sells a product through agency format and reselling format provided by a platform retailer who possesses demand forecasts from two channels.Forecast asymmetry and co-opetitive relationship arise between the platform retailer and the supplier,which affect their operational decisions and the supply chain’s performance.To improve supply chain efficiency,we compare different forecast-sharing strategies(i.e.,no forecast sharing,sharing a single forecast,and sharing two forecasts),and analyze the effects of co-opetitive parameters on the optimal forecast-sharing strategy.Our analysis shows that forecast sharing is always beneficial to the supplier,and sharing two forecasts is more beneficial than sharing a single forecast.Whereas for the platform retailer and the whole supply chain,forecast sharing is beneficial only under certain conditions,depending on the co-opetitive parameters.The optimal forecast-sharing strategy is the result of a combination of the negative effect of double marginalization in reselling channel and the positive effect of responding pricing to demand uncertainty in agency channel.We illustrate the parameter regions of the platform retailer’s voluntary sharing,contract sharing,and no sharing,and also find that higher channel competition intensity,higher market share of agency channel,and higher commission rate can promote the platform retailer’s voluntary sharing.Our study extends the research scope of demand forecast-sharing and sheds light on the decision-making processes for managing a hybrid-format online platform supply chain.

关键词： Supply chain management hybrid-format online platform demand forecast-sharing co-opetitive relationship channel competition intensity