Pub Date : 2024-09-26DOI: 10.1016/j.aml.2024.109317
Xiaojie Jing , Guirong Liu , Zhen Jin
In this paper, a Markovian SIR model on a heterogeneous network is considered. The law of large numbers and the central limit theorem of the epidemic process in a large population are provided. Further, the asymptotic distribution of the final size is given. Finally, by numerical and stochastic simulations, it is clear to show that our method performs well in approximation.
本文考虑了异构网络上的马尔可夫 SIR 模型。本文提供了大数定律和大群体中流行过程的中心极限定理。此外，还给出了最终规模的渐近分布。最后，通过数值模拟和随机模拟，可以清楚地看到我们的方法在近似方面表现良好。
{"title":"Asymptotic distribution of the final size of a stochastic SIR epidemic on heterogeneous networks","authors":"Xiaojie Jing , Guirong Liu , Zhen Jin","doi":"10.1016/j.aml.2024.109317","DOIUrl":"10.1016/j.aml.2024.109317","url":null,"abstract":"<div><div>In this paper, a Markovian SIR model on a heterogeneous network is considered. The law of large numbers and the central limit theorem of the epidemic process in a large population are provided. Further, the asymptotic distribution of the final size is given. Finally, by numerical and stochastic simulations, it is clear to show that our method performs well in approximation.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109317"},"PeriodicalIF":2.9,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142329710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-26DOI: 10.1016/j.aml.2024.109318
Xianpeng Mao , Yuning Yang
T-product based tensor principal component analysis (-tPCA) was used for dimensionality reduction, data preprocessing, compression, and visualization of multivariate data. However, -tPCA may amplify the influence of outliers and large-magnitude noise. To explore robustness against heavily corrupted third-order data, we consider the -norm tPCA model (-tPCA). We develop an effective proximal alternating maximization method and prove that within finitely many steps, the algorithm stops at a point satisfying certain optimality conditions. Numerical experiments on color face reconstruction and recognition demonstrate the efficiency of the proposed algorithms, confirming that -tPCA is more resilient to outliers compared to -tPCA.
{"title":"T-product based ℓ1-norm tensor principal component analysis and a finite-step convergence algorithm","authors":"Xianpeng Mao , Yuning Yang","doi":"10.1016/j.aml.2024.109318","DOIUrl":"10.1016/j.aml.2024.109318","url":null,"abstract":"<div><div>T-product based tensor principal component analysis (<span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-tPCA) was used for dimensionality reduction, data preprocessing, compression, and visualization of multivariate data. However, <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-tPCA may amplify the influence of outliers and large-magnitude noise. To explore robustness against heavily corrupted third-order data, we consider the <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-norm tPCA model (<span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-tPCA). We develop an effective proximal alternating maximization method and prove that within finitely many steps, the algorithm stops at a point satisfying certain optimality conditions. Numerical experiments on color face reconstruction and recognition demonstrate the efficiency of the proposed algorithms, confirming that <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>-tPCA is more resilient to outliers compared to <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-tPCA.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109318"},"PeriodicalIF":2.9,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142327558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-26DOI: 10.1016/j.aml.2024.109321
Zhiyu Fan, Hui Qi, Jing Guo
This study proposes an exact analytical approach for investigating the steady-state and transient wave dynamic propagation characteristics in frequency and time domain of rectangular sealed plate with a circular surface crack under anti-plane point source wave dynamic load. By introducing revised factor, a modified multi-directional iterative mirroring method is proposed to address the partial differential governing equations of wave propagation with boundary value conditions. Based on wave function expansion method, the scattering wave function is derived after decoupling the governing equation. Fourier integral expansion method is used to solve the infinite linear algebraic boundary equation composed of boundary value conditions. The accuracy of analytical method is verified by numerical calculation and finite element simulation. The results show that the sealed coupled waves have significant effects on dynamic stress concentration and abrupt displacement change.
{"title":"A modified multi-directional iterative mirroring method for SH waves propagation in rectangular sealed plate with a circular surface crack","authors":"Zhiyu Fan, Hui Qi, Jing Guo","doi":"10.1016/j.aml.2024.109321","DOIUrl":"10.1016/j.aml.2024.109321","url":null,"abstract":"<div><div>This study proposes an exact analytical approach for investigating the steady-state and transient wave dynamic propagation characteristics in frequency and time domain of rectangular sealed plate with a circular surface crack under anti-plane point source wave dynamic load. By introducing revised factor, a modified multi-directional iterative mirroring method is proposed to address the partial differential governing equations of wave propagation with boundary value conditions. Based on wave function expansion method, the scattering wave function is derived after decoupling the governing equation. Fourier integral expansion method is used to solve the infinite linear algebraic boundary equation composed of boundary value conditions. The accuracy of analytical method is verified by numerical calculation and finite element simulation. The results show that the sealed coupled waves have significant effects on dynamic stress concentration and abrupt displacement change.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109321"},"PeriodicalIF":2.9,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142428466","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-23DOI: 10.1016/j.aml.2024.109313
Zhaohua Li, Zhonglong Zhao
In this paper, the -periodic wave solutions of the supersymmetric KdV equation are studied by combining the super Hirota bilinear form with the super Riemann-theta function, which can be used to describe new phenomena on super quasi-periodic waves with the fermionic field. With the aid of the Gauss–Newton method, the three-periodic and four-periodic wave solutions are obtained. In particular, these quasi-periodic waves can produce parallel, crossed and degenerated patterns. The analytical method related to the characteristic lines is used to analyze the dynamic characteristics of the three-periodic and four-periodic waves. In addition, it has been indicated that -periodic waves can exist in the supersymmetric integrable systems.
本文通过超 Hirota 双线性形式与超黎曼-θ 函数的结合，研究了 N=2 超对称 KdV 方程的 N 周期波解，可用于描述费米子场超准周期波的新现象。借助高斯-牛顿方法，得到了三周期波和四周期波的解。特别是，这些准周期波可以产生平行、交叉和退化模式。与特征线相关的分析方法用于分析三周期波和四周期波的动态特性。此外，研究还指出超对称可积分系统中可能存在 N 周期波。
{"title":"N-periodic wave solutions of the N=2 supersymmetric KdV equation","authors":"Zhaohua Li, Zhonglong Zhao","doi":"10.1016/j.aml.2024.109313","DOIUrl":"10.1016/j.aml.2024.109313","url":null,"abstract":"<div><div>In this paper, the <span><math><mi>N</mi></math></span>-periodic wave solutions of the <span><math><mrow><mi>N</mi><mo>=</mo><mn>2</mn></mrow></math></span> supersymmetric KdV equation are studied by combining the super Hirota bilinear form with the super Riemann-theta function, which can be used to describe new phenomena on super quasi-periodic waves with the fermionic field. With the aid of the Gauss–Newton method, the three-periodic and four-periodic wave solutions are obtained. In particular, these quasi-periodic waves can produce parallel, crossed and degenerated patterns. The analytical method related to the characteristic lines is used to analyze the dynamic characteristics of the three-periodic and four-periodic waves. In addition, it has been indicated that <span><math><mi>N</mi></math></span>-periodic waves can exist in the supersymmetric integrable systems.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109313"},"PeriodicalIF":2.9,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142322453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-23DOI: 10.1016/j.aml.2024.109316
Nan Wang, Binbin Jiang, Meng Li
In this work, we develop high-order convex splitting implicit–explicit Runge–Kutta methods for Molecular Beam Epitaxy (MBE) model with slope selection, which plays key roles in materials science and physics for describing various phenomena, such as phase transitions, interactions and interfacial dynamics. Since the epitaxy surface height evolution equation is viewed as a dynamical form of a -gradient flow, MBE has the highly similar growing processes as the growing facets in phase-ordering process in magnetic systems. Within this context, we focus our attention on the systems with multiple components possess greater physical significance than their classical (single-component) counterparts. We rigorously prove that the proposed schemes both preserve the energy dissipation and mass conservation. Finally, the accuracy and efficiency of proposed schemes are demonstrated by some numerical experiments.
{"title":"A high-order energy stable method for the MBE models with slope selection by using Lagrange multiplier approach","authors":"Nan Wang, Binbin Jiang, Meng Li","doi":"10.1016/j.aml.2024.109316","DOIUrl":"10.1016/j.aml.2024.109316","url":null,"abstract":"<div><div>In this work, we develop high-order convex splitting implicit–explicit Runge–Kutta methods for Molecular Beam Epitaxy (MBE) model with slope selection, which plays key roles in materials science and physics for describing various phenomena, such as phase transitions, interactions and interfacial dynamics. Since the epitaxy surface height evolution equation is viewed as a dynamical form of a <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-gradient flow, MBE has the highly similar growing processes as the growing facets in phase-ordering process in magnetic systems. Within this context, we focus our attention on the systems with multiple components possess greater physical significance than their classical (single-component) counterparts. We rigorously prove that the proposed schemes both preserve the energy dissipation and mass conservation. Finally, the accuracy and efficiency of proposed schemes are demonstrated by some numerical experiments.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109316"},"PeriodicalIF":2.9,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142320207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-21DOI: 10.1016/j.aml.2024.109315
J.J. Pollacco , R.E. Baker , P.K. Maini
Real-world cellular invasion processes often take place in curved geometries. Such problems are frequently simplified in models to neglect the curved geometry in favour of computational simplicity, yet doing so risks inaccuracies in any model-based predictions. To quantify the conditions under which neglecting a curved geometry is justifiable, we explore the dynamics of a system of reaction–diffusion equations (RDEs) on a two-dimensional annular geometry analytically. Defining as the ratio of the annulus thickness and radius we derive, through an asymptotic expansion, the conditions under which it is appropriate to ignore the domain curvature for a general system of reaction–diffusion equations. To highlight the consequences of these results, we simulate solutions to the Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) model, a paradigm nonlinear RDE typically used to model spatial invasion, on an annular geometry. Thus, we quantify the size of the deviation from an analogous simulation on the rectangle, and how this deviation changes across the width of the annulus. We further characterise the nature of the solutions through numerical simulations for different values of and . Our results provide insight into when it is appropriate to neglect the domain curvature in studying travelling wave behaviour in RDEs.
{"title":"Modelling collective invasion with reaction–diffusion equations: When does domain curvature matter?","authors":"J.J. Pollacco , R.E. Baker , P.K. Maini","doi":"10.1016/j.aml.2024.109315","DOIUrl":"10.1016/j.aml.2024.109315","url":null,"abstract":"<div><div>Real-world cellular invasion processes often take place in curved geometries. Such problems are frequently simplified in models to neglect the curved geometry in favour of computational simplicity, yet doing so risks inaccuracies in any model-based predictions. To quantify the conditions under which neglecting a curved geometry is justifiable, we explore the dynamics of a system of reaction–diffusion equations (RDEs) on a two-dimensional annular geometry analytically. Defining <span><math><mi>ϵ</mi></math></span> as the ratio of the annulus thickness <span><math><mi>δ</mi></math></span> and radius <span><math><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> we derive, through an asymptotic expansion, the conditions under which it is appropriate to ignore the domain curvature for a general system of reaction–diffusion equations. To highlight the consequences of these results, we simulate solutions to the Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) model, a paradigm nonlinear RDE typically used to model spatial invasion, on an annular geometry. Thus, we quantify the size of the deviation from an analogous simulation on the rectangle, and how this deviation changes across the width of the annulus. We further characterise the nature of the solutions through numerical simulations for different values of <span><math><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> and <span><math><mi>δ</mi></math></span>. Our results provide insight into when it is appropriate to neglect the domain curvature in studying travelling wave behaviour in RDEs.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109315"},"PeriodicalIF":2.9,"publicationDate":"2024-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142357970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-19DOI: 10.1016/j.aml.2024.109309
Jan Lorenz, Tom Zwerschke, Michael Günther, Kevin Schäfers
Operator splitting methods tailored to coupled linear port-Hamiltonian systems are developed. We present algorithms that are able to exploit scalar coupling, as well as multirate potential of these coupled systems. The obtained algorithms preserve the dissipative structure of the overall system and are convergent of second order. Numerical results for coupled mass–spring–damper chains illustrate the computational efficiency of the splitting methods compared to a straight-forward application of the implicit midpoint rule to the overall system.
{"title":"Operator splitting for coupled linear port-Hamiltonian systems","authors":"Jan Lorenz, Tom Zwerschke, Michael Günther, Kevin Schäfers","doi":"10.1016/j.aml.2024.109309","DOIUrl":"10.1016/j.aml.2024.109309","url":null,"abstract":"<div><p>Operator splitting methods tailored to coupled linear port-Hamiltonian systems are developed. We present algorithms that are able to exploit scalar coupling, as well as multirate potential of these coupled systems. The obtained algorithms preserve the dissipative structure of the overall system and are convergent of second order. Numerical results for coupled mass–spring–damper chains illustrate the computational efficiency of the splitting methods compared to a straight-forward application of the implicit midpoint rule to the overall system.</p></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109309"},"PeriodicalIF":2.9,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S089396592400329X/pdfft?md5=461caa9d621b164ff71cc86835d28bc7&pid=1-s2.0-S089396592400329X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142270639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-19DOI: 10.1016/j.aml.2024.109314
Shuzhi Liu , Deqin Qiu
In this paper, the hodograph equivalent short pulse (HESP) equations are investigated via the Darboux transformation, we derive the soliton and positon solutions from the “seed” solutions, and then, the decomposition of the lower-order positons into single-solitons is given analytically when time is sufficiently large. As a notable new result, we obtain the exploding soliton and positon solutions of the sine–Gordon (SG) equation from the hodograph equivalent short pulse equations.
本文通过达尔布变换研究了霍多图等效短脉冲（HESP）方程，我们从 "种子 "解中推导出了孤子和正子解，然后，当时间 T 足够大时，分析给出了低阶正子分解为单孤子的过程。作为一项引人注目的新成果，我们从霍多图等效短脉冲方程中得到了正弦-戈登（SG）方程的爆炸孤子和正子解。
{"title":"The exploding solitons of the sine–Gordon equation","authors":"Shuzhi Liu , Deqin Qiu","doi":"10.1016/j.aml.2024.109314","DOIUrl":"10.1016/j.aml.2024.109314","url":null,"abstract":"<div><div>In this paper, the hodograph equivalent short pulse (HESP) equations are investigated via the Darboux transformation, we derive the soliton and positon solutions from the “seed” solutions, and then, the decomposition of the lower-order positons into single-solitons is given analytically when time <span><math><mi>T</mi></math></span> is sufficiently large. As a notable new result, we obtain the exploding soliton and positon solutions of the sine–Gordon (SG) equation from the hodograph equivalent short pulse equations.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109314"},"PeriodicalIF":2.9,"publicationDate":"2024-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-16DOI: 10.1016/j.aml.2024.109312
Wenjie Li , Liuan Yang , Jinde Cao
In this paper, we consider an incubation period host–pathogen system with degenerated diffusion. The global compact attractor of the solution of the model is investigated using the -contraction method. Furthermore, the basic reproduction number is defined, and we discuss the dynamic analysis of a degenerated diffusion model. The obtained theoretical results are nontrivial and can be considered a continuation of the work by Wang et al. in 2022.
本文考虑了一个具有退化扩散的潜伏期宿主-病原体系统。利用κ-收缩法研究了该模型解的全局紧凑吸引子。此外，我们还定义了基本繁殖数，并讨论了退化扩散模型的动态分析。所获得的理论结果并不复杂，可以认为是 Wang 等人 2022 年工作的延续。
{"title":"Threshold dynamics of a degenerated diffusive incubation period host–pathogen model with saturation incidence rate","authors":"Wenjie Li , Liuan Yang , Jinde Cao","doi":"10.1016/j.aml.2024.109312","DOIUrl":"10.1016/j.aml.2024.109312","url":null,"abstract":"<div><p>In this paper, we consider an incubation period host–pathogen system with degenerated diffusion. The global compact attractor of the solution of the model is investigated using the <span><math><mi>κ</mi></math></span>-contraction method. Furthermore, the basic reproduction number is defined, and we discuss the dynamic analysis of a degenerated diffusion model. The obtained theoretical results are nontrivial and can be considered a continuation of the work by Wang et al. in 2022.</p></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109312"},"PeriodicalIF":2.9,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142243781","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-13DOI: 10.1016/j.aml.2024.109310
Xiao-Tian Gao, Bo Tian
Researchers have recently become interested in a (3+1)-dimensional extended shallow water wave equation in a river or an ocean. This equation could be used to represent a variety of the physical phenomena that have some impacts on the environment, such as the floods and tsunamis. Two sets of the similarity reductions are discovered for that equation based on the variable coefficients, connected with the potential applications of the equation in a river or an ocean.
{"title":"In a river or an ocean: Similarity-reduction work on a (3+1)-dimensional extended shallow water wave equation","authors":"Xiao-Tian Gao, Bo Tian","doi":"10.1016/j.aml.2024.109310","DOIUrl":"10.1016/j.aml.2024.109310","url":null,"abstract":"<div><div>Researchers have recently become interested in a (3+1)-dimensional extended shallow water wave equation in a river or an ocean. This equation could be used to represent a variety of the physical phenomena that have some impacts on the environment, such as the floods and tsunamis. Two sets of the similarity reductions are discovered for that equation based on the variable coefficients, connected with the potential applications of the equation in a river or an ocean.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"160 ","pages":"Article 109310"},"PeriodicalIF":2.9,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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