Pub Date : 2024-04-04DOI: 10.1186/s13660-024-03131-3
Tahar Bouali, Rafik Guefaifia, Salah Boulaaras
This work deals with the existence and multiplicity of solutions for a class of variable-exponent equations involving the Kirchhoff term in variable-exponent Sobolev spaces according to some conditions, where we used the sub-supersolutions method combined with the mountain pass theory.
{"title":"Multiplicity of solutions for fractional (p ( z ) )-Kirchhoff-type equation","authors":"Tahar Bouali, Rafik Guefaifia, Salah Boulaaras","doi":"10.1186/s13660-024-03131-3","DOIUrl":"https://doi.org/10.1186/s13660-024-03131-3","url":null,"abstract":"This work deals with the existence and multiplicity of solutions for a class of variable-exponent equations involving the Kirchhoff term in variable-exponent Sobolev spaces according to some conditions, where we used the sub-supersolutions method combined with the mountain pass theory.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"147 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575075","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}
Pub Date : 2024-04-03DOI: 10.1186/s13660-024-03116-2
Abbas Najati, Yavar Khedmati Yengejeh, Kandhasamy Tamilvanan, Masho Jima Kabeto
In this article, with simple and short proofs without applying fixed point theorems, some hyperstability results corresponding to the functional equations of Cauchy and Jensen are presented in 2-normed spaces. We also obtain some results on hyperstability for the general linear functional equation $f(ax+by)=Af(x)+Bf(y)+C$ .
{"title":"Hyperstability of Cauchy and Jensen functional equations in 2-normed spaces","authors":"Abbas Najati, Yavar Khedmati Yengejeh, Kandhasamy Tamilvanan, Masho Jima Kabeto","doi":"10.1186/s13660-024-03116-2","DOIUrl":"https://doi.org/10.1186/s13660-024-03116-2","url":null,"abstract":"In this article, with simple and short proofs without applying fixed point theorems, some hyperstability results corresponding to the functional equations of Cauchy and Jensen are presented in 2-normed spaces. We also obtain some results on hyperstability for the general linear functional equation $f(ax+by)=Af(x)+Bf(y)+C$ .","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"4 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575073","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}
Pub Date : 2024-04-03DOI: 10.1186/s13660-024-03086-5
Qun Wang, Aixia Qian
We study the following nonlinear mass supercritical Kirchhoff equation: $$ - biggl(a+b int _{mathbb{R}^{N}} vert nabla u vert ^{2} biggr) triangle u+ mu u=f(u) quad text{in } {mathbb{R}^{N}}, $$ where $a ,b,m>0$ are prescribed, and the normalized constrain $int _{mathbb{R}^{N}}|u|^{2},dx =m$ is satisfied in the case $1leq Nleq 3$ . The nonlinearity f is more general and satisfies weak mass supercritical conditions. Under some mild assumptions, we establish the existence of ground state when $1leq Nleq 3$ and obtain infinitely many radial solutions when $2leq Nleq 3$ by constructing a particular bounded Palais–Smale sequence.
我们研究了以下非线性质量超临界基尔霍夫方程：$$ - biggl(a+b int _{mathbb{R}^{N}} vert nabla u vert ^{2} biggr) triangle u+ mu u=f(u) quad text{in }。{mathbb{R}^{N}}, $$ 其中$a,b,m>0$是规定的，并且归一化约束$int _mathbb{R}^{N}}|u|^{2},dx =m$在$1leq Nleq 3$的情况下是满足的。非线性 f 更为一般，满足弱质量超临界条件。在一些温和的假设条件下，我们确定了当 $1leq Nleq 3$ 时基态的存在，并通过构造一个特殊的有界 Palais-Smale 序列得到了当 $2leq Nleq 3$ 时的无穷多个径向解。
{"title":"Ground state normalized solutions to the Kirchhoff equation with general nonlinearities: mass supercritical case","authors":"Qun Wang, Aixia Qian","doi":"10.1186/s13660-024-03086-5","DOIUrl":"https://doi.org/10.1186/s13660-024-03086-5","url":null,"abstract":"We study the following nonlinear mass supercritical Kirchhoff equation: $$ - biggl(a+b int _{mathbb{R}^{N}} vert nabla u vert ^{2} biggr) triangle u+ mu u=f(u) quad text{in } {mathbb{R}^{N}}, $$ where $a ,b,m>0$ are prescribed, and the normalized constrain $int _{mathbb{R}^{N}}|u|^{2},dx =m$ is satisfied in the case $1leq Nleq 3$ . The nonlinearity f is more general and satisfies weak mass supercritical conditions. Under some mild assumptions, we establish the existence of ground state when $1leq Nleq 3$ and obtain infinitely many radial solutions when $2leq Nleq 3$ by constructing a particular bounded Palais–Smale sequence.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"107 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575276","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}
Pub Date : 2024-04-02DOI: 10.1186/s13660-024-03130-4
Fatih Hezenci, Hüseyin Budak
In this paper, we prove an equality for twice-differentiable convex functions involving the conformable fractional integrals. Moreover, several Bullen-type inequalities are established for twice-differentiable functions. More precisely, conformable fractional integrals are used to derive such inequalities. Furthermore, sundry significant inequalities are obtained by taking advantage of the convexity, Hölder inequality, and power-mean inequality. Finally, we provide our results by using special cases of obtained theorems.
{"title":"Bullen-type inequalities for twice-differentiable functions by using conformable fractional integrals","authors":"Fatih Hezenci, Hüseyin Budak","doi":"10.1186/s13660-024-03130-4","DOIUrl":"https://doi.org/10.1186/s13660-024-03130-4","url":null,"abstract":"In this paper, we prove an equality for twice-differentiable convex functions involving the conformable fractional integrals. Moreover, several Bullen-type inequalities are established for twice-differentiable functions. More precisely, conformable fractional integrals are used to derive such inequalities. Furthermore, sundry significant inequalities are obtained by taking advantage of the convexity, Hölder inequality, and power-mean inequality. Finally, we provide our results by using special cases of obtained theorems.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"18 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575170","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}
Pub Date : 2024-04-02DOI: 10.1186/s13660-024-03114-4
Hari Mohan Srivastava, Pishtiwan Othman Sabir, Sevtap Sümer Eker, Abbas Kareem Wanas, Pshtiwan Othman Mohammed, Nejmeddine Chorfi, Dumitru Baleanu
The Ruscheweyh derivative operator is used in this paper to introduce and investigate interesting general subclasses of the function class $Sigma_{m}$ of m-fold symmetric bi-univalent analytic functions. Estimates of the initial Taylor-Maclaurin coefficients $vert a_{m+1} vert $ and $vert a_{2 m+1} vert $ are obtained for functions of the subclasses introduced in this study, and the consequences of the results are discussed. Additionally, the Fekete-Szegö inequalities for these classes are investigated. The results presented could generalize and improve some recent and earlier works. In some cases, our estimates are better than the existing coefficient bounds. Furthermore, within the engineering domain, the utilization of the Ruscheweyh derivative operator can encompass a broad spectrum of engineering applications, including the robotic manipulation control, optimizing optical systems, antenna array signal processing, image compression, and control system filter design. It emphasizes the potential for innovative solutions that can significantly enhance the reliability and effectiveness of engineering applications.
{"title":"Some m-fold symmetric bi-univalent function classes and their associated Taylor-Maclaurin coefficient bounds","authors":"Hari Mohan Srivastava, Pishtiwan Othman Sabir, Sevtap Sümer Eker, Abbas Kareem Wanas, Pshtiwan Othman Mohammed, Nejmeddine Chorfi, Dumitru Baleanu","doi":"10.1186/s13660-024-03114-4","DOIUrl":"https://doi.org/10.1186/s13660-024-03114-4","url":null,"abstract":"The Ruscheweyh derivative operator is used in this paper to introduce and investigate interesting general subclasses of the function class $Sigma_{m}$ of m-fold symmetric bi-univalent analytic functions. Estimates of the initial Taylor-Maclaurin coefficients $vert a_{m+1} vert $ and $vert a_{2 m+1} vert $ are obtained for functions of the subclasses introduced in this study, and the consequences of the results are discussed. Additionally, the Fekete-Szegö inequalities for these classes are investigated. The results presented could generalize and improve some recent and earlier works. In some cases, our estimates are better than the existing coefficient bounds. Furthermore, within the engineering domain, the utilization of the Ruscheweyh derivative operator can encompass a broad spectrum of engineering applications, including the robotic manipulation control, optimizing optical systems, antenna array signal processing, image compression, and control system filter design. It emphasizes the potential for innovative solutions that can significantly enhance the reliability and effectiveness of engineering applications.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"107 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574974","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}
Pub Date : 2024-04-02DOI: 10.1186/s13660-024-03111-7
K. M. Shadimetov, J. R. Davronov
The discrete analog of the differential operator plays a significant role in constructing interpolation, quadrature, and cubature formulas. In this work, we consider a discrete analog $D_{m}(hbeta )$ of the differential operator $frac{d^{2m}}{dx^{2m}}+1$ designed specifically for even natural numbers m. The operator’s effectiveness in constructing an optimal quadrature formula in the $L_{2}^{(2,0)}(0,1)$ space is demonstrated. The errors of the optimal quadrature formula in the $W_{2}^{(2,1)}(0,1)$ space and in the $L_{2}^{(2,0)}(0,1)$ space are compared numerically. The numerical results indicate that the optimal quadrature formula constructed in this work has a smaller error than the one constructed in the $W_{2}^{(2,1)}(0,1)$ space.
{"title":"The discrete analogue of high-order differential operator and its application to finding coefficients of optimal quadrature formulas","authors":"K. M. Shadimetov, J. R. Davronov","doi":"10.1186/s13660-024-03111-7","DOIUrl":"https://doi.org/10.1186/s13660-024-03111-7","url":null,"abstract":"The discrete analog of the differential operator plays a significant role in constructing interpolation, quadrature, and cubature formulas. In this work, we consider a discrete analog $D_{m}(hbeta )$ of the differential operator $frac{d^{2m}}{dx^{2m}}+1$ designed specifically for even natural numbers m. The operator’s effectiveness in constructing an optimal quadrature formula in the $L_{2}^{(2,0)}(0,1)$ space is demonstrated. The errors of the optimal quadrature formula in the $W_{2}^{(2,1)}(0,1)$ space and in the $L_{2}^{(2,0)}(0,1)$ space are compared numerically. The numerical results indicate that the optimal quadrature formula constructed in this work has a smaller error than the one constructed in the $W_{2}^{(2,1)}(0,1)$ space.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"75 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140574973","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}
Pub Date : 2024-04-02DOI: 10.1186/s13660-024-03129-x
Yen-Chang Huang
The classical Cauchy surface area formula states that the surface area of the boundary $partial K=Sigma $ of any n-dimensional convex body in the n-dimensional Euclidean space $mathbb{R}^{n}$ can be obtained by the average of the projected areas of Σ along all directions in $mathbb{S}^{n-1}$ . In this note, we generalize the formula to the boundary of arbitrary n-dimensional submanifold in $mathbb{R}^{n}$ by introducing a natural notion of projected areas along any direction in $mathbb{S}^{n-1}$ . This surface area formula derived from the new notion coincides with not only the result of the Crofton formula but also with that of De Jong (Math. Semesterber. 60(1):81–83, 2013) by using a tubular neighborhood. We also define the projected r-volumes of Σ onto any r-dimensional subspaces and obtain a recursive formula for mean projected r-volumes of Σ.
经典的 Cauchy 表面积公式指出，n 维欧几里得空间 $mathbb{R}^{n}$ 中任意 n 维凸体的边界 $partial K=Sigma $ 的表面积可以通过 Σ 沿 $mathbb{S}^{n-1}$ 中所有方向的投影面积的平均值得到。在本注释中，我们通过引入沿 $mathbb{S}^{n-1}$ 中任意方向的投影面积的自然概念，将该公式推广到 $mathbb{R}^{n}$ 中任意 n 维子曲面的边界。这个由新概念导出的表面积公式不仅与 Crofton 公式的结果相吻合，而且与 De Jong (Math. Semesterber.Semesterber.60(1):81-83,2013）使用管状邻域的结果相吻合。我们还定义了Σ在任意r维子空间上的投影r卷，并得到了Σ的平均投影r卷的递推公式。
{"title":"A surface area formula for compact hypersurfaces in (mathbb{R}^{n})","authors":"Yen-Chang Huang","doi":"10.1186/s13660-024-03129-x","DOIUrl":"https://doi.org/10.1186/s13660-024-03129-x","url":null,"abstract":"The classical Cauchy surface area formula states that the surface area of the boundary $partial K=Sigma $ of any n-dimensional convex body in the n-dimensional Euclidean space $mathbb{R}^{n}$ can be obtained by the average of the projected areas of Σ along all directions in $mathbb{S}^{n-1}$ . In this note, we generalize the formula to the boundary of arbitrary n-dimensional submanifold in $mathbb{R}^{n}$ by introducing a natural notion of projected areas along any direction in $mathbb{S}^{n-1}$ . This surface area formula derived from the new notion coincides with not only the result of the Crofton formula but also with that of De Jong (Math. Semesterber. 60(1):81–83, 2013) by using a tubular neighborhood. We also define the projected r-volumes of Σ onto any r-dimensional subspaces and obtain a recursive formula for mean projected r-volumes of Σ.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"42 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140575070","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}
Pub Date : 2024-03-29DOI: 10.1186/s13660-024-03126-0
Eungil Ko, Ji Eun Lee, Jongrak Lee
In this paper, we study several properties of an orthonormal basis ${N_{n}(z)}$ for the Newton space $N^{2}({mathbb{P}})$ . In particular, we investigate the product of $N_{m}$ and $N_{m}$ and the orthogonal projection P of $overline{N_{n}}N_{m}$ that maps from $L^{2}(mathbb{P})$ onto $N^{2}(mathbb{P})$ . Moreover, we find the matrix representation of Toeplitz operators with respect to such an orthonormal basis on the Newton space $N^{2}({mathbb{P}})$ .
{"title":"Matrix representation of Toeplitz operators on Newton spaces","authors":"Eungil Ko, Ji Eun Lee, Jongrak Lee","doi":"10.1186/s13660-024-03126-0","DOIUrl":"https://doi.org/10.1186/s13660-024-03126-0","url":null,"abstract":"In this paper, we study several properties of an orthonormal basis ${N_{n}(z)}$ for the Newton space $N^{2}({mathbb{P}})$ . In particular, we investigate the product of $N_{m}$ and $N_{m}$ and the orthogonal projection P of $overline{N_{n}}N_{m}$ that maps from $L^{2}(mathbb{P})$ onto $N^{2}(mathbb{P})$ . Moreover, we find the matrix representation of Toeplitz operators with respect to such an orthonormal basis on the Newton space $N^{2}({mathbb{P}})$ .","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"54 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140325350","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}
Pub Date : 2024-03-27DOI: 10.1186/s13660-024-03123-3
Satit Saejung
We explore the intermixed method for finding a common element of the intersection of the solution set of a mixed variational inequality and the fixed point set of a nonexpansive mapping. We point out that Khuangsatung and Kangtunyakarn’s statement [J. Inequal. Appl. 2023:1, 2023] regarding the resolvent utilized in their paper is not correct. To resolve this gap, we employ the epigraphical projection and the product space approach. In particular, we obtain a strong convergence theorem with a weaker assumption.
{"title":"On the intermixed method for mixed variational inequality problems: another look and some corrections","authors":"Satit Saejung","doi":"10.1186/s13660-024-03123-3","DOIUrl":"https://doi.org/10.1186/s13660-024-03123-3","url":null,"abstract":"We explore the intermixed method for finding a common element of the intersection of the solution set of a mixed variational inequality and the fixed point set of a nonexpansive mapping. We point out that Khuangsatung and Kangtunyakarn’s statement [J. Inequal. Appl. 2023:1, 2023] regarding the resolvent utilized in their paper is not correct. To resolve this gap, we employ the epigraphical projection and the product space approach. In particular, we obtain a strong convergence theorem with a weaker assumption.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"104 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140317063","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}
Pub Date : 2024-03-26DOI: 10.1186/s13660-024-03125-1
Mehran Ghaderi, Shahram Rezapour
Recent research indicates the need for improved models of physical phenomena with multiple shocks. One of the newest methods is to use differential inclusions instead of differential equations. In this work, we intend to investigate the existence of solutions for an m-dimensional system of quantum differential inclusions. To ensure the existence of the solution of inclusions, researchers typically rely on the Arzela–Ascoli and Nadler’s fixed point theorems. However, we have taken a different approach and utilized the endpoint technique of the fixed point theory to guarantee the solution’s existence. This sets us apart from other researchers who have used different methods. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables, and some figures. The paper ends with an example.
最近的研究表明，需要改进具有多重冲击的物理现象模型。最新的方法之一是用微分夹杂代替微分方程。在这项工作中，我们打算研究 m 维量子微分夹杂系统解的存在性。为了确保夹杂解的存在性，研究人员通常依赖 Arzela-Ascoli 和 Nadler 定点定理。然而，我们采取了不同的方法，利用定点理论的端点技术来保证解的存在性。这使我们有别于其他使用不同方法的研究者。为了更好地理解问题和验证结果，我们给出了数值算法、表格和一些图表。论文以一个实例结束。
{"title":"On an m-dimensional system of quantum inclusions by a new computational approach and heatmap","authors":"Mehran Ghaderi, Shahram Rezapour","doi":"10.1186/s13660-024-03125-1","DOIUrl":"https://doi.org/10.1186/s13660-024-03125-1","url":null,"abstract":"Recent research indicates the need for improved models of physical phenomena with multiple shocks. One of the newest methods is to use differential inclusions instead of differential equations. In this work, we intend to investigate the existence of solutions for an m-dimensional system of quantum differential inclusions. To ensure the existence of the solution of inclusions, researchers typically rely on the Arzela–Ascoli and Nadler’s fixed point theorems. However, we have taken a different approach and utilized the endpoint technique of the fixed point theory to guarantee the solution’s existence. This sets us apart from other researchers who have used different methods. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables, and some figures. The paper ends with an example.","PeriodicalId":16088,"journal":{"name":"Journal of Inequalities and Applications","volume":"44 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315380","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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