The aim of this paper is to provide new representations and computations of the generalized Moore-Penrose inverse. Based on the Moore-Penrose inverse, group inverse, Bott-Duffin inverse and certain projections, some representations for the generalized Moore-Penrose inverse are given. An equivalent condition for the continuity of the generalized Moore-Penrose inverse is proposed. Splitting methods and successive matrix squaring algorithm for computing the generalized Moore-Penrose inverse are presented.
{"title":"Further representations and computations of the generalized Moore-Penrose inverse","authors":"Kezheng Zuo, Yang Chen, Li Yuan","doi":"10.3934/math.20231191","DOIUrl":"https://doi.org/10.3934/math.20231191","url":null,"abstract":"The aim of this paper is to provide new representations and computations of the generalized Moore-Penrose inverse. Based on the Moore-Penrose inverse, group inverse, Bott-Duffin inverse and certain projections, some representations for the generalized Moore-Penrose inverse are given. An equivalent condition for the continuity of the generalized Moore-Penrose inverse is proposed. Splitting methods and successive matrix squaring algorithm for computing the generalized Moore-Penrose inverse are presented.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":"1 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70162999","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the Riccati matrix equation $ X A^{-1} X = B $ in which the conventional matrix products are generalized to the semi-tensor products $ ltimes $. When $ A $ and $ B $ are positive definite matrices satisfying the factor-dimension condition, this equation has a unique positive definite solution, which is defined to be the metric geometric mean of $ A $ and $ B $. We show that this geometric mean is the maximum solution of the Riccati inequality. We then extend the notion of the metric geometric mean to positive semidefinite matrices by a continuity argument and investigate its algebraic properties, order properties and analytic properties. Moreover, we establish some equations and inequalities of metric geometric means for matrices involving cancellability, positive linear map and concavity. Our results generalize the conventional metric geometric means of matrices.
研究了Riccati矩阵方程$ X A^{-1} X = B $，其中常规矩阵积推广为半张量积$ ltimes $。当$ A $和$ B $为满足因子维条件的正定矩阵时，该方程有一个唯一的正定解，定义为$ A $和$ B $的度量几何平均值。我们证明了这个几何平均值是里卡蒂不等式的最大解。然后通过连续性论证将度量几何均值的概念推广到正半定矩阵，并研究了它的代数性质、阶性质和解析性质。此外，我们还建立了包含可消性、正线性映射和凹性的矩阵的度量几何均值的方程和不等式。我们的结果推广了矩阵的常规度量几何平均值。
{"title":"Riccati equation and metric geometric means of positive semidefinite matrices involving semi-tensor products","authors":"P. Chansangiam, Arnon Ploymukda","doi":"10.3934/math.20231195","DOIUrl":"https://doi.org/10.3934/math.20231195","url":null,"abstract":"We investigate the Riccati matrix equation $ X A^{-1} X = B $ in which the conventional matrix products are generalized to the semi-tensor products $ ltimes $. When $ A $ and $ B $ are positive definite matrices satisfying the factor-dimension condition, this equation has a unique positive definite solution, which is defined to be the metric geometric mean of $ A $ and $ B $. We show that this geometric mean is the maximum solution of the Riccati inequality. We then extend the notion of the metric geometric mean to positive semidefinite matrices by a continuity argument and investigate its algebraic properties, order properties and analytic properties. Moreover, we establish some equations and inequalities of metric geometric means for matrices involving cancellability, positive linear map and concavity. Our results generalize the conventional metric geometric means of matrices.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":"1 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70163285","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abeer O. Badghaish, Abdel Moneim Y. Lashin, Amani Z. Bajamal, Fayzah A. Alshehri
In this paper, we introduce a new subclass of analytic and bi-univalent functions in the open unit disc $ U. $ For this subclass of functions, estimates of the initial coefficients $ leftvert A_{2}rightvert $ and $ leftvert A_{3}rightvert $ of the Taylor-Maclaurin series are given. An application of Legendre polynomials to this subclass of functions is presented. Furthermore, our study discusses several special cases.
{"title":"A new subclass of analytic and bi-univalent functions associated with Legendre polynomials","authors":"Abeer O. Badghaish, Abdel Moneim Y. Lashin, Amani Z. Bajamal, Fayzah A. Alshehri","doi":"10.3934/math.20231196","DOIUrl":"https://doi.org/10.3934/math.20231196","url":null,"abstract":"In this paper, we introduce a new subclass of analytic and bi-univalent functions in the open unit disc $ U. $ For this subclass of functions, estimates of the initial coefficients $ leftvert A_{2}rightvert $ and $ leftvert A_{3}rightvert $ of the Taylor-Maclaurin series are given. An application of Legendre polynomials to this subclass of functions is presented. Furthermore, our study discusses several special cases.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":"42 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70163671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Leyla Sağ Dönmez, Abdurrahman Büyükkaya, M. Öztürk
In this study, we characterize a novel contraction mapping referred to as $ alpha_{i}^{j} $-$ left({bf D}_{{mathscr{C}}}left(mathfrak{P}_{hat E}right)right) $-contraction in light of $ {bf D}_{mathscr{C}} $-contraction mappings associated with the Geraghty-type contraction and $ E $-type contraction. Besides, a novel common fixed-point theorem providing such mappings is demonstrated in the context of partial $ flat $-metric spaces. It is stated that the main theorem is a generalization of the existing literature, and its comparisons with the results are expressed. Additionally, the efficiency of the result of this study is demonstrated through some examples and an application to homotopy theory.
{"title":"Fixed-point results via $ alpha_{i}^{j} $-$ left({bf D}_{{mathscr{C}}}left(mathfrak{P}_{hat E}right)right) $-contractions in partial $ flat $-metric spaces","authors":"Leyla Sağ Dönmez, Abdurrahman Büyükkaya, M. Öztürk","doi":"10.3934/math.20231204","DOIUrl":"https://doi.org/10.3934/math.20231204","url":null,"abstract":"In this study, we characterize a novel contraction mapping referred to as $ alpha_{i}^{j} $-$ left({bf D}_{{mathscr{C}}}left(mathfrak{P}_{hat E}right)right) $-contraction in light of $ {bf D}_{mathscr{C}} $-contraction mappings associated with the Geraghty-type contraction and $ E $-type contraction. Besides, a novel common fixed-point theorem providing such mappings is demonstrated in the context of partial $ flat $-metric spaces. It is stated that the main theorem is a generalization of the existing literature, and its comparisons with the results are expressed. Additionally, the efficiency of the result of this study is demonstrated through some examples and an application to homotopy theory.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":"1 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70163768","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, a new error bound for the linear complementarity problems of $ B- $matrices which is a subclass of the $ P- $matrices is presented. Theoretical analysis and numerical example illustrate that the new error bound improves some existing results.
{"title":"A new error bound for linear complementarity problems involving $ B- $matrices","authors":"Hongmin Mo, Yingxue Dong","doi":"10.3934/math.20231218","DOIUrl":"https://doi.org/10.3934/math.20231218","url":null,"abstract":"In this paper, a new error bound for the linear complementarity problems of $ B- $matrices which is a subclass of the $ P- $matrices is presented. Theoretical analysis and numerical example illustrate that the new error bound improves some existing results.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":"1 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70164314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, the Tsallis and Renyi extropy is presented as a continuous measure of information under the continuous distribution. Furthermore, the features and their connection to other information measures are introduced. Some stochastic comparisons and results on the order statistics and upper records are given. Moreover, some theorems about the maximum Tsallis and Renyi extropy are discussed. On the other hand, numerical results of the non-parametric estimation of Tsallis extropy are calculated for simulated and real data with application to time series model and its forecasting.
{"title":"Continuous Tsallis and Renyi extropy with pharmaceutical market application","authors":"M. Mohamed, Najwan Alsadat, O. S. Balogun","doi":"10.3934/math.20231233","DOIUrl":"https://doi.org/10.3934/math.20231233","url":null,"abstract":"In this paper, the Tsallis and Renyi extropy is presented as a continuous measure of information under the continuous distribution. Furthermore, the features and their connection to other information measures are introduced. Some stochastic comparisons and results on the order statistics and upper records are given. Moreover, some theorems about the maximum Tsallis and Renyi extropy are discussed. On the other hand, numerical results of the non-parametric estimation of Tsallis extropy are calculated for simulated and real data with application to time series model and its forecasting.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":"1 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70164915","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper investigates an unreliable $ M/G(P_{1}, P_{2})/1 $ retrial queueing system with a woking vacation. An arriving customer successfully starts the first phase service with the probability $ alpha $ or the server fails with the probability $ bar{alpha} $. Once failure happens, the serving customer is taken to the orbit. The failed server is taken for repair with some delay. Once the repair is comleted, the server is ready to provide service once again. In this background, we implemented the working vacation scenario. During working vacation, the service will be provided at a slower rate, rather than entirely stopping the service. The supplementary variable method was adopted to find the orbit and system lengths. Additionally, some unique results and numerical evaluations have been presented.
{"title":"Unreliable retrial queueing system with working vacation","authors":"Bharathy Shanmugam, M. C. Saravanarajan","doi":"10.3934/math.20231234","DOIUrl":"https://doi.org/10.3934/math.20231234","url":null,"abstract":"This paper investigates an unreliable $ M/G(P_{1}, P_{2})/1 $ retrial queueing system with a woking vacation. An arriving customer successfully starts the first phase service with the probability $ alpha $ or the server fails with the probability $ bar{alpha} $. Once failure happens, the serving customer is taken to the orbit. The failed server is taken for repair with some delay. Once the repair is comleted, the server is ready to provide service once again. In this background, we implemented the working vacation scenario. During working vacation, the service will be provided at a slower rate, rather than entirely stopping the service. The supplementary variable method was adopted to find the orbit and system lengths. Additionally, some unique results and numerical evaluations have been presented.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":"1 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70165316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we establish a mass formula for self-orthogonal codes, quasi self-dual codes, and self-dual codes over commutative non-unital rings $ {{mathit {I}_p}} = left < a, b | pa = pb = 0, a^2 = b, ab = 0 right > $, where $ p $ is an odd prime. We also give a classification of the three said classes of codes over $ {{mathit {I}_p}} $ where $ p = 3, 5, $ and $ 7 $, with lengths up to $ 3 $.
本文建立了可交换非一元环$ {{mathit {I}_p}} = 左< a, b | pa = pb = 0, a^2 = b, ab = 0 右> $上的自正交码、拟自对偶码和自对偶码的质量公式，其中$ p $为奇素数。我们还给出了$ {mathit {I}_p}} $上的三种代码类的分类，其中$ p = 3,5，$和$ 7 $，长度最多为$ 3 $。
{"title":"The mass formula for self-orthogonal and self-dual codes over a non-unitary commutative ring","authors":"A. Alahmadi, A. Alshuhail, P. Solé","doi":"10.3934/math.20231242","DOIUrl":"https://doi.org/10.3934/math.20231242","url":null,"abstract":"In this paper, we establish a mass formula for self-orthogonal codes, quasi self-dual codes, and self-dual codes over commutative non-unital rings $ {{mathit {I}_p}} = left < a, b | pa = pb = 0, a^2 = b, ab = 0 right > $, where $ p $ is an odd prime. We also give a classification of the three said classes of codes over $ {{mathit {I}_p}} $ where $ p = 3, 5, $ and $ 7 $, with lengths up to $ 3 $.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":"1 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70165454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Rashad Ismail, S. Hameed, Uzma Ahmad, Khadija Majeed, M. Javaid
For a signature function $ Psi:E({H}) longrightarrow {pm 1} $ with underlying graph $ H $, a signed graph (S.G) $ hat{H} = (H, Psi) $ is a graph in which edges are assigned the signs using the signature function $ Psi $. An S.G $ hat{H} $ is said to fulfill the symmetric eigenvalue property if for every eigenvalue $ hat{h}(hat{H}) $ of $ hat{H} $, $ -hat{h}(hat{H}) $ is also an eigenvalue of $ hat{H} $. A non singular S.G $ hat{H} $ is said to fulfill the property $ (mathcal{SR}) $ if for every eigenvalue $ hat{h}(hat{H}) $ of $ hat{H} $, its reciprocal is also an eigenvalue of $ hat{H} $ (with multiplicity as that of $ hat{h}(hat{H}) $). A non singular S.G $ hat{H} $ is said to fulfill the property $ (-mathcal{SR}) $ if for every eigenvalue $ hat{h}(hat{H}) $ of $ hat{H} $, its negative reciprocal is also an eigenvalue of $ hat{H} $ (with multiplicity as that of $ hat{h}(hat{H}) $). In this article, non bipartite unbalanced S.Gs $ hat{mathfrak{C}}^{(m, 1)}_{3} $ and $ hat{mathfrak{C}}^{(m, 2)}_{5} $, where $ m $ is even positive integer have been constructed and it has been shown that these graphs fulfill the symmetric eigenvalue property, the S.Gs $ hat{mathfrak{C}}^{(m, 1)}_{3} $ also fulfill the properties $ (-mathcal{SR}) $ and $ (mathcal{SR}) $, whereas the S.Gs $ hat{mathfrak{C}}^{(m, 2)}_{5} $ are close to fulfill the properties $ (-mathcal{SR}) $ and $ (mathcal{SR}) $.
{"title":"Unbalanced signed graphs with eigenvalue properties","authors":"Rashad Ismail, S. Hameed, Uzma Ahmad, Khadija Majeed, M. Javaid","doi":"10.3934/math.20231262","DOIUrl":"https://doi.org/10.3934/math.20231262","url":null,"abstract":"For a signature function $ Psi:E({H}) longrightarrow {pm 1} $ with underlying graph $ H $, a signed graph (S.G) $ hat{H} = (H, Psi) $ is a graph in which edges are assigned the signs using the signature function $ Psi $. An S.G $ hat{H} $ is said to fulfill the symmetric eigenvalue property if for every eigenvalue $ hat{h}(hat{H}) $ of $ hat{H} $, $ -hat{h}(hat{H}) $ is also an eigenvalue of $ hat{H} $. A non singular S.G $ hat{H} $ is said to fulfill the property $ (mathcal{SR}) $ if for every eigenvalue $ hat{h}(hat{H}) $ of $ hat{H} $, its reciprocal is also an eigenvalue of $ hat{H} $ (with multiplicity as that of $ hat{h}(hat{H}) $). A non singular S.G $ hat{H} $ is said to fulfill the property $ (-mathcal{SR}) $ if for every eigenvalue $ hat{h}(hat{H}) $ of $ hat{H} $, its negative reciprocal is also an eigenvalue of $ hat{H} $ (with multiplicity as that of $ hat{h}(hat{H}) $). In this article, non bipartite unbalanced S.Gs $ hat{mathfrak{C}}^{(m, 1)}_{3} $ and $ hat{mathfrak{C}}^{(m, 2)}_{5} $, where $ m $ is even positive integer have been constructed and it has been shown that these graphs fulfill the symmetric eigenvalue property, the S.Gs $ hat{mathfrak{C}}^{(m, 1)}_{3} $ also fulfill the properties $ (-mathcal{SR}) $ and $ (mathcal{SR}) $, whereas the S.Gs $ hat{mathfrak{C}}^{(m, 2)}_{5} $ are close to fulfill the properties $ (-mathcal{SR}) $ and $ (mathcal{SR}) $.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":"1 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70165991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
At present, the reliability of interconnection networks of multiprocessing systems has become a hot topic of research concern for parallel computer systems. Conditional connectivity is an important parameter to measure the reliability of an interconnected network. In reality, the failure of one node will inevitably have a negative impact on the surrounding nodes. Often it is the specific structures that fail in an interconnected network. Therefore, we propose two novel kinds of connectivity, called $ g $-extra $ H $-structure connectivity and $ g $-extra $ H $-substructure connectivity, to go for a more accurate measure of the reliability of the network. Hypercube network is the most dominant interconnection network topology used by computer systems today, for example, the famous parallel computing systems Cray $ T3D $, Cray $ T3E $, $ IBM $ Blue Gene, etc. are built with it as the interconnection network topology. In this paper, we obtain the results of the $ g $-extra $ H $-structure connectivity and the $ g $-extra $ H $-substructure connectivity of the hypercubes when the specific structure is $ P_k $ and $ g = 1 $.
目前，多处理系统互连网络的可靠性问题已成为并行计算机系统研究的热点问题。条件连通性是衡量互联网络可靠性的重要参数。在现实中，一个节点的故障不可避免地会对周围的节点产生负面影响。通常是特定的结构在互联网络中失效。因此，我们提出了两种新的连接，称为$ g $-额外$ H $-结构连接和$ g $-额外$ H $-子结构连接，以更准确地衡量网络的可靠性。超立方体网络是当今计算机系统使用的最主流的互联网络拓扑，例如著名的并行计算系统Cray $ T3D $、Cray $ T3E $、IBM $ Blue Gene等都是以超立方体网络作为互联网络拓扑构建的。本文得到了超立方体在特定结构为$ P_k $和$ g = 1 $时的$ g $-extra $ H $-结构连通性和$ g $-extra $ H $-子结构连通性的结果。
{"title":"The $ g $-extra $ H $-structure connectivity and $ g $-extra $ H $-substructure connectivity of hypercubes","authors":"Bo Zhu, Shumin Zhang, Huifen Ge, Chengfu Ye","doi":"10.3934/math.20231267","DOIUrl":"https://doi.org/10.3934/math.20231267","url":null,"abstract":"At present, the reliability of interconnection networks of multiprocessing systems has become a hot topic of research concern for parallel computer systems. Conditional connectivity is an important parameter to measure the reliability of an interconnected network. In reality, the failure of one node will inevitably have a negative impact on the surrounding nodes. Often it is the specific structures that fail in an interconnected network. Therefore, we propose two novel kinds of connectivity, called $ g $-extra $ H $-structure connectivity and $ g $-extra $ H $-substructure connectivity, to go for a more accurate measure of the reliability of the network. Hypercube network is the most dominant interconnection network topology used by computer systems today, for example, the famous parallel computing systems Cray $ T3D $, Cray $ T3E $, $ IBM $ Blue Gene, etc. are built with it as the interconnection network topology. In this paper, we obtain the results of the $ g $-extra $ H $-structure connectivity and the $ g $-extra $ H $-substructure connectivity of the hypercubes when the specific structure is $ P_k $ and $ g = 1 $.","PeriodicalId":48562,"journal":{"name":"AIMS Mathematics","volume":"1 1","pages":""},"PeriodicalIF":2.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70166766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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