Applied Mathematics and Computation最新文献

{"title":"Integrated mobile facility production and distribution scheduling planning; A synchronized solution framework","authors":"Reza Ramezanian ,&nbsp;Mohammad Hosein Mahdavi ,&nbsp;Shahrooz Shahparvari","doi":"10.1016/j.amc.2025.129277","DOIUrl":"10.1016/j.amc.2025.129277","url":null,"abstract":"<div><div>This research explores supply chain modeling for mobile production facilities, emphasizing distribution and manufacturing processes. The focus is on a fleet of production-equipped trucks that manufacture orders during transit to customer locations. A mixed-integer linear programming (MILP) model is proposed to address the production and delivery planning problem, minimizing total costs, including production, transportation, and driver expenses. The computational strategy integrates Benders Decomposition with a Rolling Horizon heuristic. The accelerated Benders Decomposition reduced computational time by 25–35 % across large-scale scenarios. Numerical experiments demonstrated that the hybrid method achieved optimal or near-optimal solutions with cost savings of 12–18 % compared to standalone methods. For example, in scenarios with 15 mobile facilities and 10 products, the methodology outperformed CPLEX in solution time and quality. The results highlight the practical applicability of the proposed framework, enabling e-commerce enterprises to enhance customer service levels by reducing delivery times by 10–15 %. This study provides a robust and efficient approach to optimizing synchronized production and delivery operations while eliminating the costs associated with fixed manufacturing facilities.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"494 ","pages":"Article 129277"},"PeriodicalIF":3.5,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990383","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}

{"title":"A new mathematical model and solution method for the asymmetric traveling salesman problem with replenishment arcs","authors":"K. Gulnaz Bulbul ,&nbsp;Refail Kasimbeyli","doi":"10.1016/j.amc.2025.129278","DOIUrl":"10.1016/j.amc.2025.129278","url":null,"abstract":"<div><div>This paper presents a new mathematical model for the Asymmetric Traveling Salesman Problem with Replenishment Arcs, an extension of the Asymmetric Traveling Salesman Problem, incorporating constraints on subpaths within the tour. Many existing modeling approaches to this problem require the generation of replenishment feasible or replenishment violation paths as a parameter set, which may lead to computational difficulties. Our formulation addresses these difficulties and provides direct computation of an optimal tour without relying on the set of paths as a parameter set. In this paper, we also propose a Lagrangian relaxation-based solution method. Given that ordinary Lagrangian functions can encounter duality gap in nonconvex problems, we employ a special augmented Lagrangian function, which is proven to overcome the issue of duality gap for many classes of nonconvex problems, including ours. In this paper, we utilize a hybrid solution method by combining the F-MSG method with an ant colony optimization algorithm. A similar solution method was previously used in Bulbul and Kasimbeyli (2021) <span><span>[13]</span></span>. In this paper, the method used in the aforementioned paper is enhanced in terms of computational complexity and solution efficiency. We assess the proposed method on 180 randomly generated instances, demonstrating that it achieves optimal solutions for almost all cases. Additionally, we apply our methodology to the aircraft maintenance routing problem, testing it on 11 instances from the aforementioned study. The results highlight the effectiveness of our approach, with an average improvement of 48.6% in solution time and a 0.93% enhancement in solution quality.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"494 ","pages":"Article 129278"},"PeriodicalIF":3.5,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990386","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}

{"title":"Comparative analysis of nonlinear Urysohn functional integral equations via Nyström method","authors":"Imtiyaz Ahmad Bhat ,&nbsp;Lakshmi Narayan Mishra ,&nbsp;Vishnu Narayan Mishra","doi":"10.1016/j.amc.2025.129287","DOIUrl":"10.1016/j.amc.2025.129287","url":null,"abstract":"<div><div>In this sequel, the investigation of a diverse range of Urysohn type nonlinear functional integral equations falling under the Fredholm type is examined. This equation encompasses various integral and functional equations as specific instances. By imposing smoothness conditions on the involved functions, both the solution's existence and its uniqueness using the fixed point method are established. Subsequently, we employ the Nyström method for solution's approximation, leading to a set of algebraic equations of nonlinear form. The Picard iterative method is then applied to the solution's approximation for this system of algebraic equations. Additionally, the trapezoidal method is applied to approximate the solution and a novel Grönwall inequality is used to establish the convergence of method, providing a reliable theoretical foundation. Numerical examples and a comparative analysis are presented to demonstrate the convergence, effectiveness, and superiority of the Nyström method compared to the trapezoidal method, highlighting its improved practicality and versatility.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"494 ","pages":"Article 129287"},"PeriodicalIF":3.5,"publicationDate":"2025-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990382","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}

{"title":"Natural extension of a Lie algebra A4,18b,|b|≤1, realizations, invariant systems and integrability","authors":"Muhammad Ayub ,&nbsp;Saira Bano","doi":"10.1016/j.amc.2025.129274","DOIUrl":"10.1016/j.amc.2025.129274","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In the frame of Lie theory, the classical successive reduction of order for scalar ODEs is not much effective for system of differential equations due to technical hurdles. Lie algebraic approach is utilized to overcome these hurdles, then further invoked for the classification, linearization and integrability of system of differential equations. But in the case of higher dimension, this approach has also its own limitations due to classification of low dimensional Lie algebras and corresponding realizations. In the realm of Lie algebra theory, a natural extension refers to the process that involves enlarging a Lie algebra by adding new elements or structures while maintaining the fundamental algebraic properties. This extension technique finds applications in various fields, including physics, geometry, and mathematics.&lt;/div&gt;&lt;div&gt;In this paper, retaining &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;18&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; as a base algebra in &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-dimensional space, examined in &lt;span&gt;&lt;span&gt;[2]&lt;/span&gt;&lt;/span&gt;, a possible natural extension of underlying Lie algebra is investigated and also constructed associated realizations. Lie algebraic approach with natural extension is utilized for the analysis of a system of &lt;em&gt;n&lt;/em&gt; second-order ODEs possessing extended form of Lie algebra &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;18&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; in &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-dimensional space to &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-dimensional space. In this research work, the extension of underlying Lie algebra, corresponding realizations and canonical forms associated with systems under consideration, have been formulated. Moreover, the integrability of investigated canonical forms for system of &lt;em&gt;n&lt;/em&gt; second-order ODEs affiliated with extended form of Lie algebra &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;18&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;|&lt;/mo&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; in &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-dimensional space to &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-dimensional space is described and three integrable classes of underlying systems have been deduced. In addition, two new classes of algebraic linearization criteria for a system of &lt;em&gt;n&lt;/em&gt; second-order ODEs admitting Lie algebra &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;18&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"494 ","pages":"Article 129274"},"PeriodicalIF":3.5,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990369","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}

{"title":"New stabilized mixed finite element methods for two-field poroelasticity with low permeability","authors":"Linshuang He,&nbsp;Luru Jing,&nbsp;Minfu Feng","doi":"10.1016/j.amc.2025.129285","DOIUrl":"10.1016/j.amc.2025.129285","url":null,"abstract":"<div><div>In this paper, we develop new stabilized mixed finite element (MFE) methods for two-field Biot's model of poroelasticity. We employ the H(div)-conforming element and discontinuous element to approximate the displacement and pressure variables, and use the <em>θ</em>-scheme to discretize time. By adding the stabilization term based on polynomial pressure projection, the fully-discrete and stabilized MFE methods are obtained. Our methods work well for both inf-sup stable and unstable element pairs, and provide oscillation-free pressure solutions in heterogeneous materials with low-permeable layers or interfaces. These methods are also volumetric locking-free and locally mass-conservative. We establish optimal <em>a priori</em> error estimates and perform numerical examples, which show the uniform robustness of the proposed methods for low permeability.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"494 ","pages":"Article 129285"},"PeriodicalIF":3.5,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990280","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}

{"title":"Convergence analysis of an efficient multistep pseudo-spectral continuous Galerkin approach for solving Volterra integro-differential equations","authors":"Yin Yang ,&nbsp;Pai Yao ,&nbsp;Emran Tohidi","doi":"10.1016/j.amc.2025.129284","DOIUrl":"10.1016/j.amc.2025.129284","url":null,"abstract":"<div><div>In this research article, we apply the multistep pseudo-spectral continuous Galerkin approach for solving the first-order Volterra integro-differential equations by the aid of the orthogonal Legendre polynomials. This approach is a recursive scheme that the accuracy of the numerical solution at the present subinterval depends on the numerical solutions at the previous subintervals. Convergence analysis of the suggested numerical approach is discussed via using an important auxiliary problem. Extensive numerical test problems with high oscillating analytical solutions, exact solutions with steep gradients, and long time computational intervals are considered and both of the h-version and p-version convergence rates are examined experimentally. Finally, the conclusions regarding the presented approach and the considered model are provided and we point out to some other models that can be solved numerically via this efficient and robust method.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"494 ","pages":"Article 129284"},"PeriodicalIF":3.5,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990368","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}

{"title":"A composite filter-based approach to adaptive prescribed-time output-feedback control of strict-feedback nonlinear systems with output and input quantization","authors":"Chen Chen,&nbsp;Jinghao Li","doi":"10.1016/j.amc.2025.129279","DOIUrl":"10.1016/j.amc.2025.129279","url":null,"abstract":"<div><div>This paper investigates the adaptive prescribed-time control problems of strict-feedback nonlinear systems with output quantization, input quantization and unknown control coefficients. Firstly, a quantized K-filter is constructed to estimate the system states. Then, a set of command filters are introduced to smooth the discontinuous stabilizing functions and reduce the computational burden. Moreover, a class of scaling functions are designed to achieve the desired prescribed-time tracking performance. Based on the quantized K-filter, a set of command filters and a class of scaling functions, a set of new error variables are established to facilitate the design of the quantized composite filter-based adaptive prescribed-time control method. It is proved that the tracking error converges to an adjustable compact set within a prescribed time. Finally, two examples are provided to illustrate the effectiveness of the proposed method.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"494 ","pages":"Article 129279"},"PeriodicalIF":3.5,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990367","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}

{"title":"Data-driven bipartite synchronization control of multi-agent systems with asymmetric input saturation over switching networks","authors":"Ziwen Shen ,&nbsp;Tao Dong ,&nbsp;Tingwen Huang","doi":"10.1016/j.amc.2025.129280","DOIUrl":"10.1016/j.amc.2025.129280","url":null,"abstract":"<div><div>This paper addresses the bipartite synchronization (BS) issue of nonlinear discrete-time multi-agent systems (NDTMASs) with asymmetric input saturation in switching cooperative-competitive networks. First, a local neighborhood error system (LNES) is constructed based on the switching cooperative-competitive networks. By utilizing a hyperbolic function, we design a novel local cost function and derive an optimal control policy. Based on this control policy, we construct a novel saturation-input policy iteration (SIPI) technique. Furthermore, we establish the convergence of the SIPI algorithm. It is demonstrated that the LNES can converge to zero, and thus the BS problem for MASs can be solved. To implement the SIPI algorithm, a control framework based on an actor-critic architecture is constructed. To achieve asymmetric input saturation, we select an asymmetric constrained function as the activation function for the actor network. Finally, an example is presented to illustrate the reliability of our approach.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"494 ","pages":"Article 129280"},"PeriodicalIF":3.5,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990366","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}

{"title":"Stochastic stability of networked control systems with packet disordering via a gain-scheduling controller","authors":"Huihui Ji ,&nbsp;Shengyuan Xu ,&nbsp;He Zhang ,&nbsp;Chenlong Li ,&nbsp;Yuhua Xu","doi":"10.1016/j.amc.2025.129282","DOIUrl":"10.1016/j.amc.2025.129282","url":null,"abstract":"<div><div>It is addressed the problem of almost sure stability for networked control systems (NCSs), where the order of the control signal is possible to be disordering over a communication network. A stochastic configuration network algorithm is employed to record and predict the packet order, which is random to allot input biases or weights and evaluates the output wights. Under the algorithm, a stochastic parameter appears in the controlled system and is assumed to obey a Markov distribution. In this paper, a group of scheduling controllers in a controlled interval are chosen subjected to an underlying Markov chain and the execution time allotted to the controller is subjected to a hidden stochastic process. Thus, the NCSs based on stochastic scheduling include complex random variables. Combining Lyapunov analysis method and some stochastic analysis tools, almost sure stability conditions are given. Finally, the obtained results are applied into addressing the problem of stochastic scheduling control of a quadruple-tank process.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"494 ","pages":"Article 129282"},"PeriodicalIF":3.5,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142990365","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}

{"title":"Leveraging surrounding past strategies to maintain cooperation in the perverse prisoner's dilemma","authors":"Akihiro Takahara ,&nbsp;Tomoko Sakiyama","doi":"10.1016/j.amc.2024.129272","DOIUrl":"10.1016/j.amc.2024.129272","url":null,"abstract":"<div><div>In spatial game theory, developed models preserving cooperators often incorporate memory to enhance realism. This study examined the role of memory in a spatial prisoner's dilemma. In the proposed model, all players use their own and their neighbors’ past memories and current states to update strategies under specific conditions. When a player's score is lower than that of a neighbor using the same strategy, and that neighbor has the highest score among all neighbors, the player revisits past strategies and adopts a less experienced strategy. This rule adjusts behavior under unfavorable conditions. Results showed that the proposed model effectively retains cooperators. Previous studies have often necessitated the use of long-term memory and intricate systems; however, the present model eliminates these requirements. Instead, it sustains cooperation by employing a strategy that relies on minimal historical information when determining the subsequent course of action.</div></div>","PeriodicalId":55496,"journal":{"name":"Applied Mathematics and Computation","volume":"493 ","pages":"Article 129272"},"PeriodicalIF":3.5,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142968138","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}