Alfred Göpfert, Christiane Tammer, Constantin Zălinescu
Abstract Being informed that one of our articles is cited in the paper mentioned in the title, we downloaded it, and we were surprised to see that, practically, all the results from our paper were reproduced in Section 3 of Meghea and Stamin’s article. Having in view the title of the article, one is tempted to think that the remarks mentioned in the paper are original and there are examples given as to where and how (at least) some of the reviewed results are effectively applied. Unfortunately, a closer look shows that most of those remarks in Section 3 are, in fact, extracted from our article, and it is not shown how a specific result is used in a certain application. So, our aim in the present note is to discuss the content of Section 3 of Meghea and Stamin’s paper, emphasizing their Remark 8, in which it is asserted that the proof of Lemma 7 in our article is “full of errors.”
{"title":"On I. Meghea and C. S. Stamin review article “Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,” Demonstratio Mathematica 2022; 55: 354–379","authors":"Alfred Göpfert, Christiane Tammer, Constantin Zălinescu","doi":"10.1515/dema-2023-0102","DOIUrl":"https://doi.org/10.1515/dema-2023-0102","url":null,"abstract":"Abstract Being informed that one of our articles is cited in the paper mentioned in the title, we downloaded it, and we were surprised to see that, practically, all the results from our paper were reproduced in Section 3 of Meghea and Stamin’s article. Having in view the title of the article, one is tempted to think that the remarks mentioned in the paper are original and there are examples given as to where and how (at least) some of the reviewed results are effectively applied. Unfortunately, a closer look shows that most of those remarks in Section 3 are, in fact, extracted from our article, and it is not shown how a specific result is used in a certain application. So, our aim in the present note is to discuss the content of Section 3 of Meghea and Stamin’s paper, emphasizing their Remark 8, in which it is asserted that the proof of Lemma 7 in our article is “full of errors.”","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135912979","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}
Abstract In this manuscript, we first generate topological structures by subset neighborhoods and ideals and apply to establish some generalized rough-set models. Then, we present other types of generalized rough-set models directly defined by the concepts of subset neighborhoods and ideals. We explore the main characterizations of the proposed approximation spaces and compare them in terms of approximation operators and accuracy measures. The obtained results and given examples show that the second type of the proposed approximation spaces is better than the first one in cases of u u and ⟨ u ⟩ langle urangle , whereas the relationships between the rest of the six cases are posted as an open question. Moreover, we demonstrate the advantages of the current models to decrease the upper approximation and increase the lower approximation compared to the existing approaches in published literature. Algorithms and a flow chart are given to illustrate how the exact and rough sets are determined for each approach. Finally, we analyze the information system of dengue fever to confirm the efficiency of our approaches to maximize the value of accuracy and shrink the boundary regions.
{"title":"Approximation spaces inspired by subset rough neighborhoods with applications","authors":"T. Al-shami, A. Mhemdi","doi":"10.1515/dema-2022-0223","DOIUrl":"https://doi.org/10.1515/dema-2022-0223","url":null,"abstract":"Abstract In this manuscript, we first generate topological structures by subset neighborhoods and ideals and apply to establish some generalized rough-set models. Then, we present other types of generalized rough-set models directly defined by the concepts of subset neighborhoods and ideals. We explore the main characterizations of the proposed approximation spaces and compare them in terms of approximation operators and accuracy measures. The obtained results and given examples show that the second type of the proposed approximation spaces is better than the first one in cases of u u and ⟨ u ⟩ langle urangle , whereas the relationships between the rest of the six cases are posted as an open question. Moreover, we demonstrate the advantages of the current models to decrease the upper approximation and increase the lower approximation compared to the existing approaches in published literature. Algorithms and a flow chart are given to illustrate how the exact and rough sets are determined for each approach. Finally, we analyze the information system of dengue fever to confirm the efficiency of our approaches to maximize the value of accuracy and shrink the boundary regions.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49402220","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}
S. A. Mohiuddine, Karunesh Singh, Abdullah Alotaibi
Abstract In this article, we the study generalized family of positive linear operators based on two parameters, which are a broad family of many well-known linear positive operators, e.g., Baskakov-Durrmeyer, Baskakov-Szász, Szász-Beta, Lupaş-Beta, Lupaş-Szász, genuine Bernstein-Durrmeyer, and Pǎltǎnea. We first find direct estimates in terms of the second-order modulus of continuity, then we find an estimate of error in the Ditzian-Totik modulus of smoothness. Then we discuss the rate of approximation for functions in the Lipschitz class. Furthermore, we study the pointwise Grüss-Voronovskaja-type result and also establish the Grüss-Voronovskaja-type asymptotic formula in quantitative form.
{"title":"On the order of approximation by modified summation-integral-type operators based on two parameters","authors":"S. A. Mohiuddine, Karunesh Singh, Abdullah Alotaibi","doi":"10.1515/dema-2022-0182","DOIUrl":"https://doi.org/10.1515/dema-2022-0182","url":null,"abstract":"Abstract In this article, we the study generalized family of positive linear operators based on two parameters, which are a broad family of many well-known linear positive operators, e.g., Baskakov-Durrmeyer, Baskakov-Szász, Szász-Beta, Lupaş-Beta, Lupaş-Szász, genuine Bernstein-Durrmeyer, and Pǎltǎnea. We first find direct estimates in terms of the second-order modulus of continuity, then we find an estimate of error in the Ditzian-Totik modulus of smoothness. Then we discuss the rate of approximation for functions in the Lipschitz class. Furthermore, we study the pointwise Grüss-Voronovskaja-type result and also establish the Grüss-Voronovskaja-type asymptotic formula in quantitative form.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42506467","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}
Abstract In this article, we consider compositions of positive integers with 2s and 3s. We see that these compositions lead us to results that involve Padovan numbers, and we give some tiling models of these compositions. Moreover, we examine some tiling models of the compositions related to the Padovan polynomials and prove some identities using the tiling model’s method. Next, we obtain various identities of the compositions of positive integers with 2s and 3s related to the Padovan numbers. The number of palindromic compositions of this type is determined, and some numerical arithmetic functions are defined. Finally, we provide a table that compares all of the results obtained from compositions of positive integers with 2s and 3s.
{"title":"Compositions of positive integers with 2s and 3s","authors":"Orhan Dişkaya, H. Menken","doi":"10.1515/dema-2022-0227","DOIUrl":"https://doi.org/10.1515/dema-2022-0227","url":null,"abstract":"Abstract In this article, we consider compositions of positive integers with 2s and 3s. We see that these compositions lead us to results that involve Padovan numbers, and we give some tiling models of these compositions. Moreover, we examine some tiling models of the compositions related to the Padovan polynomials and prove some identities using the tiling model’s method. Next, we obtain various identities of the compositions of positive integers with 2s and 3s related to the Padovan numbers. The number of palindromic compositions of this type is determined, and some numerical arithmetic functions are defined. Finally, we provide a table that compares all of the results obtained from compositions of positive integers with 2s and 3s.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41718598","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}
Abstract We study the uniqueness, the continuity in L 2 {L}^{2} , and the large time decay for the Leray solutions of the 3D incompressible Navier-Stokes equations with the nonlinear exponential damping term a ( e b ∣ u ∣ 2 − 1 ) u aleft({e}^{b| u{| }^{{bf{2}}}}-1)u , ( a , b > 0 a,bgt 0 ).
摘要我们研究了三维不可压缩Navier-Stokes方程Leray解的唯一性、在L2｛L｝^｛2｝中的连续性和大时间衰减，该方程具有非线性指数阻尼项a（e bÜuÜ2−1）u a left（｛e｝^｝b | u｛|｝^）u，（a，b＞0 a，b gt 0）。
{"title":"Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping","authors":"Mongi Blel, J. Benameur","doi":"10.1515/dema-2022-0208","DOIUrl":"https://doi.org/10.1515/dema-2022-0208","url":null,"abstract":"Abstract We study the uniqueness, the continuity in L 2 {L}^{2} , and the large time decay for the Leray solutions of the 3D incompressible Navier-Stokes equations with the nonlinear exponential damping term a ( e b ∣ u ∣ 2 − 1 ) u aleft({e}^{b| u{| }^{{bf{2}}}}-1)u , ( a , b > 0 a,bgt 0 ).","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49554098","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}
Abstract In this article, we investigate a class of analytic functions defined on the unit open disc U = { z : ∣ z ∣ < 1 } {mathcal{U}}=left{z:| z| lt 1right} , such that for every f ∈ P α ( β , γ ) fin {{mathcal{P}}}_{alpha }left(beta ,gamma ) , α > 0 alpha gt 0 , 0 ≤ β ≤ 1 0le beta le 1 , 0 < γ ≤ 1 0lt gamma le 1 , and ∣ z ∣ < 1 | z| lt 1 , the inequality Re f ′ ( z ) + 1 − γ α γ z f ″ ( z ) − β 1 − β > 0 {rm{Re}}left{frac{f^{prime} left(z)+frac{1-gamma }{alpha gamma }z{f}^{^{primeprime} }left(z)-beta }{1-beta }right}gt 0 holds. We find conditions on the numbers α , β alpha ,beta , and γ gamma such that P α ( β , γ ) ⊆ S P ( λ ) {{mathcal{P}}}_{alpha }left(beta ,gamma )subseteq SPleft(lambda ) , for λ ∈ ( − π 2 , π 2 ) lambda in left(-frac{pi }{2},frac{pi }{2}) , where S P ( λ ) SPleft(lambda ) denotes the set of all λ lambda -spirallike functions. We also make use of Ruscheweyh’s duality theory to derive conditions on the numbers α , β , γ alpha ,beta ,gamma and the real-valued function φ varphi so that the integral operator V φ ( f ) {V}_{varphi }(f) maps the set P α ( β , γ ) {{mathcal{P}}}_{alpha }left(beta ,gamma ) into the set S P ( λ ) SPleft(lambda ) , provided φ varphi is non-negative normalized function ( ∫ 0 1 φ ( t ) d t = 1 ) left({int }_{0}^{1}varphi left(t){rm{d}}t=1) and V φ ( f ) ( z ) = ∫ 0 1 φ ( t ) f ( t z ) t d t . {V}_{varphi }(f)left(z)=underset{0}{overset{1}{int }}varphi left(t)frac{fleft(tz)}{t}{rm{d}}t.
{"title":"Duality for convolution on subclasses of analytic functions and weighted integral operators","authors":"E. Amini, M. Fardi, S. Al-Omari, K. Nonlaopon","doi":"10.1515/dema-2022-0168","DOIUrl":"https://doi.org/10.1515/dema-2022-0168","url":null,"abstract":"Abstract In this article, we investigate a class of analytic functions defined on the unit open disc U = { z : ∣ z ∣ < 1 } {mathcal{U}}=left{z:| z| lt 1right} , such that for every f ∈ P α ( β , γ ) fin {{mathcal{P}}}_{alpha }left(beta ,gamma ) , α > 0 alpha gt 0 , 0 ≤ β ≤ 1 0le beta le 1 , 0 < γ ≤ 1 0lt gamma le 1 , and ∣ z ∣ < 1 | z| lt 1 , the inequality Re f ′ ( z ) + 1 − γ α γ z f ″ ( z ) − β 1 − β > 0 {rm{Re}}left{frac{f^{prime} left(z)+frac{1-gamma }{alpha gamma }z{f}^{^{primeprime} }left(z)-beta }{1-beta }right}gt 0 holds. We find conditions on the numbers α , β alpha ,beta , and γ gamma such that P α ( β , γ ) ⊆ S P ( λ ) {{mathcal{P}}}_{alpha }left(beta ,gamma )subseteq SPleft(lambda ) , for λ ∈ ( − π 2 , π 2 ) lambda in left(-frac{pi }{2},frac{pi }{2}) , where S P ( λ ) SPleft(lambda ) denotes the set of all λ lambda -spirallike functions. We also make use of Ruscheweyh’s duality theory to derive conditions on the numbers α , β , γ alpha ,beta ,gamma and the real-valued function φ varphi so that the integral operator V φ ( f ) {V}_{varphi }(f) maps the set P α ( β , γ ) {{mathcal{P}}}_{alpha }left(beta ,gamma ) into the set S P ( λ ) SPleft(lambda ) , provided φ varphi is non-negative normalized function ( ∫ 0 1 φ ( t ) d t = 1 ) left({int }_{0}^{1}varphi left(t){rm{d}}t=1) and V φ ( f ) ( z ) = ∫ 0 1 φ ( t ) f ( t z ) t d t . {V}_{varphi }(f)left(z)=underset{0}{overset{1}{int }}varphi left(t)frac{fleft(tz)}{t}{rm{d}}t.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48766088","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}
Abstract The harmonic numbers and generalized harmonic numbers appear frequently in many diverse areas such as combinatorial problems, many expressions involving special functions in analytic number theory, and analysis of algorithms. The aim of this article is to derive some identities involving generalized harmonic numbers and generalized harmonic functions from the beta functions F n ( x ) = B ( x + 1 , n + 1 ) , ( n = 0 , 1 , 2 , … ) {F}_{n}left(x)=Bleft(x+1,n+1),left(n=0,1,2,ldots ) using elementary methods. For instance, we show that the Hurwitz zeta function ζ ( x + 1 , r ) zeta left(x+1,r) and r ! r! are expressed in terms of those numbers and functions, for every r = 2 , 3 , 4 , 5 r=2,3,4,5 .
调和数和广义调和数经常出现在许多不同的领域，如组合问题、解析数论中许多涉及特殊函数的表达式、算法分析等。本文的目的是利用初等方法，从函数F n (x)=B (x+1,n+1)， (n=0,1,2，…){F＿n}{}left (x)=B left (x+1,n+1)， left (n=0,1,2, ldots)中导出一些涉及广义调和数和广义调和函数的恒等式。例如，我们证明了Hurwitz ζ函数ζ (x+1,r) zetaleft (x+1,r)和r !r!用这些数和函数表示，对于每个r=2,3,4,5 r=2,3,4,5。
{"title":"Some identities on generalized harmonic numbers and generalized harmonic functions","authors":"Dae San Kim, H. Kim, Taekyun Kim","doi":"10.1515/dema-2022-0229","DOIUrl":"https://doi.org/10.1515/dema-2022-0229","url":null,"abstract":"Abstract The harmonic numbers and generalized harmonic numbers appear frequently in many diverse areas such as combinatorial problems, many expressions involving special functions in analytic number theory, and analysis of algorithms. The aim of this article is to derive some identities involving generalized harmonic numbers and generalized harmonic functions from the beta functions F n ( x ) = B ( x + 1 , n + 1 ) , ( n = 0 , 1 , 2 , … ) {F}_{n}left(x)=Bleft(x+1,n+1),left(n=0,1,2,ldots ) using elementary methods. For instance, we show that the Hurwitz zeta function ζ ( x + 1 , r ) zeta left(x+1,r) and r ! r! are expressed in terms of those numbers and functions, for every r = 2 , 3 , 4 , 5 r=2,3,4,5 .","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41775430","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}
Abstract The aim of this article is to study the fully degenerate Bernoulli polynomials and numbers, which are a degenerate version of Bernoulli polynomials and numbers and arise naturally from the Volkenborn integral of the degenerate exponential functions on Z p {{mathbb{Z}}}_{p} . We find some explicit expressions for the fully degenerate Bernoulli polynomials and numbers in terms of the degenerate Stirling numbers of the second kind, the degenerate r r -Stirling numbers of the second kind, and the degenerate Stirling polynomials. We also consider the degenerate poly-Bernoulli polynomials and derive explicit representations for them in terms of the same degenerate Stirling numbers and polynomials.
{"title":"Fully degenerate Bernoulli numbers and polynomials","authors":"Taekyun Kim, Dae San Kim, Jin-Woo Park","doi":"10.1515/dema-2022-0160","DOIUrl":"https://doi.org/10.1515/dema-2022-0160","url":null,"abstract":"Abstract The aim of this article is to study the fully degenerate Bernoulli polynomials and numbers, which are a degenerate version of Bernoulli polynomials and numbers and arise naturally from the Volkenborn integral of the degenerate exponential functions on Z p {{mathbb{Z}}}_{p} . We find some explicit expressions for the fully degenerate Bernoulli polynomials and numbers in terms of the degenerate Stirling numbers of the second kind, the degenerate r r -Stirling numbers of the second kind, and the degenerate Stirling polynomials. We also consider the degenerate poly-Bernoulli polynomials and derive explicit representations for them in terms of the same degenerate Stirling numbers and polynomials.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44954720","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}
Abstract The pointwise estimates of the deviation T n , A f ( ⋅ ) − f ( ⋅ ) {T}_{n,A}f(cdot )-fleft(cdot ) in terms of pointwise moduli of continuity based on the points of differentiability of indefinite integral of f f , with application of the rth differences of the entries of A A , are proved. The similar results in case of the Lebesgue points are considered, too. Analogical results on norm approximation with remarks and corollaries are also given.
{"title":"Approximation of integrable functions by general linear matrix operators of their Fourier series","authors":"V. Mishra, W. Łenski, B. Szal","doi":"10.1515/dema-2022-0009","DOIUrl":"https://doi.org/10.1515/dema-2022-0009","url":null,"abstract":"Abstract The pointwise estimates of the deviation T n , A f ( ⋅ ) − f ( ⋅ ) {T}_{n,A}f(cdot )-fleft(cdot ) in terms of pointwise moduli of continuity based on the points of differentiability of indefinite integral of f f , with application of the rth differences of the entries of A A , are proved. The similar results in case of the Lebesgue points are considered, too. Analogical results on norm approximation with remarks and corollaries are also given.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49636939","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}
Abstract In this article, we will prove the existence of infinitely many solutions for a class of quasilinear Schrödinger equations without assuming the 4-superlinear at infinity on the nonlinearity. We achieve our goal by using the Fountain theorem.
{"title":"Infinitely many solutions for quasilinear Schrödinger equations with sign-changing nonlinearity without the aid of 4-superlinear at infinity","authors":"M. Khiddi, L. Essafi","doi":"10.1515/dema-2022-0169","DOIUrl":"https://doi.org/10.1515/dema-2022-0169","url":null,"abstract":"Abstract In this article, we will prove the existence of infinitely many solutions for a class of quasilinear Schrödinger equations without assuming the 4-superlinear at infinity on the nonlinearity. We achieve our goal by using the Fountain theorem.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42693140","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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