28 July 2025, Volume 48 Issue 4

    

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  Well-Posedness and Stability of Solutions in Set Optimization

  ZHANG CONGJUN, WANG ZHIWEI, LI SAI

  Acta Mathematicae Applicatae Sinica. 2025, 48(4): 511-528. https://doi.org/10.20142/j.cnki.amas.202501007

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  The aim of this paper is to investigate the well-posedness and stability in set optimization. For the first time, by embedding the original set optimization problem into a family of perturbed problems with the same structure, the notions of generalized $m_{1}$-well-posedness and $m_{1}$-well-setness under perturbation for set optimization problems are introduced. The relationships between them are obtained. Sufficient condition, necessary conditon and many characterizations are given for generalized $m_{1}$-well-posedness under perturbation, respectively. Two new definitions of monotonicity of set-valued mapping are introduced, and from this, the semi-continuity and closedness of $m_{1}$-efficient solution mappings for parametric set optimization problems are studied.
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  Theoretical Analysis and Approximate Calculation of Metric Dimension Problem of Graphs

  WU JIAN, WANG LI, YANG WEIHUA

  Acta Mathematicae Applicatae Sinica. 2025, 48(4): 529-551. https://doi.org/10.20142/j.cnki.amas.202501034

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  The metric dimension problem (MDP) for graphs is a class of combinatorial optimization problems widely used in machine navigation, chemistry, network discovery and other fields. In order to solve this problem, a general optimization model with combinatorial properties is established in this paper, the resolving relationships between vertices and vertex pairs are described theoretically, the resolving graph and resolving probability distribution of graphs are proposed, and then the resolving probability distributions of some special graph classes are determined. An approximate metric dimension computation framework based on reinforcement learning and graph convolutional neural network is established by using machine learning sampling method. Numerical experiments show the effectiveness of graph machine learning algorithm to solve the graph metric dimension problem.
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  A Class of Adapted Power Generalized Weibull Model for Recurrent Event Data

  SU SHANSHAN, ZHOU JIE, CHENG WENHUI

  Acta Mathematicae Applicatae Sinica. 2025, 48(4): 552-571. https://doi.org/10.20142/j.cnki.amas.202401014

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  The adapted power generalized Weibull (APGW) model, which is generalized from the Weibull distribution, is a flexible parametric model and includes many popular survival models as special cases. In this paper, we extend the APGW model to the modeling of recurrent event data. A maximum likelihood estimating approach is proposed and the asymptotic properties are established. A model selection method is provided via variable selection through the SCAD penalty likelihood approach. The performance of both parameter estimation and model selection under finite samples is examined by simulation studies. To the end, the proposed method is applied to a bladder cancer recurrent event data for illustration.
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  The Extinction and Persistence of Androgen Deprivation Therapy Model with Lévy Noise

  TAN YUANSHUN, YANG HUAN

  Acta Mathematicae Applicatae Sinica. 2025, 48(4): 572-596. https://doi.org/10.20142/j.cnki.amas.202501009

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  During the process of androgen deprivation therapy for prostate cancer (PCa), stochastic effects in the tumor microenvironment can cause treatment to fall short of the desired effect on tumor clearance. In this paper, the Lévy noise is introduced to describe the random changes of tumor cell number, and a system of stochastic differential equations (SDEs) driven by Lévy noise is established and analyzed. Firstly, through appropriate Lyapunov functions and the solution formula of SDEs driven by Lévy noise, the existence and uniqueness of the global positive solution are proved. Then, employing the theories and methods of stochastic dynamical systems, such as Lévy-Itô's formula, the comparison theorem of SDEs driven by Lévy noise, the exponential martingale inequality and the Borel-Cantelli lemma, the stochastic dynamics of extinction, non-persistence in the mean, weak persistence in the mean and stochastic permanence of PCa cells are studied. Finally, numerical simulation further verifies the theoretical results. Combined with theoretical analysis and numerical simulation, it can be found that the higher the intensity of Lévy noise, the easier the PCa cells are removed.
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  Projection Uniformity of q-level Design under Reproducing Kernel Function

  TIAN JIEZHONG, LI HONGYI, ZHANG SHIXIAN

  Acta Mathematicae Applicatae Sinica. 2025, 48(4): 597-610. https://doi.org/10.20142/j.cnki.amas.202401045

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  Uniform designs are a kind of space-filling designs that are widely used in scientific experiments and industrial production due to their runs flexibility. Discrepancies are used to measure the overall uniformity of designs, but in practical problems, it is often necessary to consider the uniformity of the low-dimensional projection designs. In this paper, the conclusions of Wang et al. (2021) are generalized. Firstly, the uniformity pattern of any $q$-level design based on the reproducing kernel function is defined, and the minimum projection uniformity criterion is provided to select the designs with minimum low-dimensional projection uniformity. Secondly, the analytical relationship between the uniformity pattern and the generalized word length pattern of $q$-level designs is built, and an improved lower bound of the uniformity pattern is obtained, which can be a basis to measure the uniformity of projection designs. Finally, some numerical examples are given to verify the theoretical results.
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  Characterization of J-Selfadjoint Extension Domains for Singular Two-interval High-order J-Symmetric Differential Operators

  FAN BEISHENG, XU MEIZHEN

  Acta Mathematicae Applicatae Sinica. 2025, 48(4): 611-629. https://doi.org/10.20142/j.cnki.amas.202501035

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  In this paper, under the assumption that the regularity domain $\Pi(T_{0})$ of the minimal operator $T_{0}$ on two intervals is non-empty, we give an analytic characterization of the $J$-self-adjoint extension domains for singular higher-order $J$-symmetric differential operators on two intervals by employing the direct sum theory and the solutions to the equation $\tau y=\lambda y$. This result includes the case when the endpoints are regular or singular and the deficiency index is arbitrary.
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  Quantum Bernoulli Noise Method for Diffusion-type Stochastic Schrödinger Equation

  REN SULING, SUN GUIXIANG, ZHAO JUNJIE

  Acta Mathematicae Applicatae Sinica. 2025, 48(4): 630-650. https://doi.org/10.20142/j.cnki.amas.202501019

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  Quantum Bernoulli noises are the family of annihilation and creation operators on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal time. Stochastic Schrödinger equation is a classical stochastic differential equation describing the evolution of open quantum systems in continuous quantum measurement. In this paper, the diffusion of linear stochastic Schrödinger equation with quantum Bernoulli noise method is studied. This model is expected to play a role in describing the evolution of open quantum systems interacting with QBN. Firstly, we prove some technical conclusions in Bernoulli functional space. In particular, we get spectral decomposition of counting operators. And then, with these theorems as main tools, inspired by the ideas of Pellegrini and Mora, we establish a theorem about the existence and uniqueness of a regular solution to the associated stochastic Schrödinger equation, which gives particular versions of previous results via QBN. Finally, we prove that the diffusion case can be obtained by the limit of the discrete process obtained by the measurement of non-diagonal observables, and some further results are obtained.
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  Traveling Waves in a Nonlocal Dispersal Seir Epidemic Model with Delay and Saturated Recovery Rate

  WEI ZHENNI, ZHAO HAIQIN

  Acta Mathematicae Applicatae Sinica. 2025, 48(4): 651-674. https://doi.org/10.20142/j.cnki.amas.202501017

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  The paper is concerned with the traveling wave solutions of a nonlocal dispersal delayed SEIR model with saturated recovery rate. First, we proved the existence of traveling wave solutions for R₀>1,c> $c^{*}$ by applying the Schauder's fixed point theorem and a limiting argument, and the asymptotic boundary of the traveling wave solutions was obtained by Lyapunov functional. Secondly, we proved the nonexistence of the traveling wave solutions for R₀>1, 0<c<c^* or R₀<1 by two-side Laplace transform. Finally, we discussed how the latent period and the movement of the infective individuals affect the minimal wave speed. Generally speaking, the longer the latent period, the slower the spread of the disease.