Pub Date : 2024-02-28DOI: 10.1007/s11117-024-01035-6
Yassine El Gantouh
The aim of this work is to provide useful criteria for well-posedness, positivity and stability of a class of infinite-dimensional linear systems. These criteria are based on an inverse estimate with respect to the Hille–Yosida Theorem. Indeed, we establish a generation result for perturbed positive operator semigroups, namely, for positive unbounded boundary perturbations. This unifies previous results available in the literature and that were established separately so far. We also prove that uniform exponential stability persists under unbounded boundary perturbations. Finally, applications to a Boltzmann equation with non-local boundary conditions on a finite network and a size-dependent population system with delayed birth process are also presented.
{"title":"Well-posedness and stability of a class of linear systems","authors":"Yassine El Gantouh","doi":"10.1007/s11117-024-01035-6","DOIUrl":"https://doi.org/10.1007/s11117-024-01035-6","url":null,"abstract":"<p>The aim of this work is to provide useful criteria for well-posedness, positivity and stability of a class of infinite-dimensional linear systems. These criteria are based on an inverse estimate with respect to the Hille–Yosida Theorem. Indeed, we establish a generation result for perturbed positive operator semigroups, namely, for positive unbounded boundary perturbations. This unifies previous results available in the literature and that were established separately so far. We also prove that uniform exponential stability persists under unbounded boundary perturbations. Finally, applications to a Boltzmann equation with non-local boundary conditions on a finite network and a size-dependent population system with delayed birth process are also presented.</p>","PeriodicalId":54596,"journal":{"name":"Positivity","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140010301","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 : 2024-02-25DOI: 10.1007/s11117-024-01033-8
Anthony W. Hager, Brian Wynne
The Yosida representation for an Archimedean vector lattice A with weak unit u, denoted (A, u), reveals similarities between the ideas of the title, FST and SMP. If A is Archimedean, the conclusion of the FST means exactly that for each (0 < e in A), the Yosida space for ((e^{dd},e)), denoted (Y_e), has a base of clopen sets. This yields a short “Yosida based" proof of FST. On the other hand, SMP implies that each (Y_e) has a (pi )-base of clopen sets. The converse fails, but holds if A has a strong unit (and in a somewhat more general situation).
具有弱单位 u 的阿基米德向量网格 A 的约西达表示(表示为 (A, u))揭示了标题、FST 和 SMP 之间的相似性。如果 A 是阿基米德的,那么 FST 的结论就意味着,对于每个 (0 < e in A), ((e^{dd},e)) 的 Yosida 空间,表示为 (Y_e),有一个开集的基。这就得到了一个简短的 "基于 Yosida "的 FST 证明。另一方面,SMP 意味着每个 (Y_e) 都有一个开集的基(pi )。反之亦然,但如果 A 有一个强单元则成立(在更一般的情况下)。
{"title":"The Freudenthal spectral theorem and sufficiently many projections in Archimedean vector lattices","authors":"Anthony W. Hager, Brian Wynne","doi":"10.1007/s11117-024-01033-8","DOIUrl":"https://doi.org/10.1007/s11117-024-01033-8","url":null,"abstract":"<p>The Yosida representation for an Archimedean vector lattice <i>A</i> with weak unit <i>u</i>, denoted (<i>A</i>, <i>u</i>), reveals similarities between the ideas of the title, FST and SMP. If <i>A</i> is Archimedean, the conclusion of the FST means exactly that for each <span>(0 < e in A)</span>, the Yosida space for <span>((e^{dd},e))</span>, denoted <span>(Y_e)</span>, has a base of clopen sets. This yields a short “Yosida based\" proof of FST. On the other hand, SMP implies that each <span>(Y_e)</span> has a <span>(pi )</span>-base of clopen sets. The converse fails, but holds if <i>A</i> has a strong unit (and in a somewhat more general situation).</p>","PeriodicalId":54596,"journal":{"name":"Positivity","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139977708","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 : 2024-02-23DOI: 10.1007/s11117-024-01034-7
Sorin G. Gal, Constantin P. Niculescu
In this paper we extend Korovkin’s theorem to the context of sequences of weakly nonlinear and monotone operators defined on certain Banach function spaces. Several examples illustrating the theory are included.
{"title":"Nonlinear operator extensions of Korovkin’s theorems","authors":"Sorin G. Gal, Constantin P. Niculescu","doi":"10.1007/s11117-024-01034-7","DOIUrl":"https://doi.org/10.1007/s11117-024-01034-7","url":null,"abstract":"<p>In this paper we extend Korovkin’s theorem to the context of sequences of weakly nonlinear and monotone operators defined on certain Banach function spaces. Several examples illustrating the theory are included.\u0000</p>","PeriodicalId":54596,"journal":{"name":"Positivity","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139947380","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 : 2024-02-21DOI: 10.1007/s11117-024-01031-w
Rovshan Bandaliyev, Dunya Aliyeva
In this paper we give necessary and sufficient conditions for the boundedness of the multidimensional Hausdorff operator on weighted Lebesgue spaces. In particular, we establish necessary and sufficient conditions for the boundedness of special type of the multidimensional Hausdorff operator on weighted Lebesgue spaces for monotone radial weight functions. Also, we get similar results for important operators of harmonic analysis which are special cases of the multidimensional Hausdorff operator. The results are illustrated by a number of examples.
{"title":"A characterization of two-weight norm inequalities for multidimensional Hausdorff operators on Lebesgue spaces","authors":"Rovshan Bandaliyev, Dunya Aliyeva","doi":"10.1007/s11117-024-01031-w","DOIUrl":"https://doi.org/10.1007/s11117-024-01031-w","url":null,"abstract":"<p>In this paper we give necessary and sufficient conditions for the boundedness of the multidimensional Hausdorff operator on weighted Lebesgue spaces. In particular, we establish necessary and sufficient conditions for the boundedness of special type of the multidimensional Hausdorff operator on weighted Lebesgue spaces for monotone radial weight functions. Also, we get similar results for important operators of harmonic analysis which are special cases of the multidimensional Hausdorff operator. The results are illustrated by a number of examples.</p>","PeriodicalId":54596,"journal":{"name":"Positivity","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139927141","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 : 2024-02-07DOI: 10.1007/s11117-024-01029-4
Abstract
In this work, we study a comparison of norms in non-commutative spaces of (tau )-measurable operators associated with a semifinite von Neumann algebra. In particular, we obtain Nazarov–Podkorytov type lemma Nazarov et al. (Complex analysis, operators, and related topics. Operatory theory: advances, vol 113, pp 247–267, 2000) and extend the main results in Astashkin et al. (Math Ann, 2023. https://doi.org/10.1007/s00208-023-02606-w) to non-commutative settings. Moreover, we complete the range of the parameter p for (0<p<1.)
摘要 在这项工作中,我们研究了与(tau )半有限 von Neumann 代数相关的可测算子的非交换空间中的规范比较。特别是,我们得到了 Nazarov-Podkorytov 型 Lemma Nazarov 等人 (Complex analysis, operators, and related topics.运算理论:进展,第 113 卷,第 247-267 页,2000 年),并将阿斯塔什金等人(Math Ann, 2023. https://doi.org/10.1007/s00208-023-02606-w)的主要结果扩展到非交换环境。此外,我们完成了参数 p 的范围,即 (0<p<1.)
{"title":"Extensions of Nazarov–Podkorytov lemma in non-commutative spaces of $$tau $$ -measurable operators","authors":"","doi":"10.1007/s11117-024-01029-4","DOIUrl":"https://doi.org/10.1007/s11117-024-01029-4","url":null,"abstract":"<h3>Abstract</h3> <p>In this work, we study a comparison of norms in non-commutative spaces of <span> <span>(tau )</span> </span>-measurable operators associated with a semifinite von Neumann algebra. In particular, we obtain Nazarov–Podkorytov type lemma Nazarov et al. (Complex analysis, operators, and related topics. Operatory theory: advances, vol 113, pp 247–267, 2000) and extend the main results in Astashkin et al. (Math Ann, 2023. <span>https://doi.org/10.1007/s00208-023-02606-w</span>) to non-commutative settings. Moreover, we complete the range of the parameter <em>p</em> for <span> <span>(0<p<1.)</span> </span></p>","PeriodicalId":54596,"journal":{"name":"Positivity","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139771917","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 : 2024-01-28DOI: 10.1007/s11117-023-01027-y
Kazimierz Musiał
Let ((X, {{mathfrak {A}}},P)) and ((Y, {{mathfrak {B}}},Q)) be two probability spaces and R be their skew product on the product (sigma )-algebra ({{mathfrak {A}}}otimes {{mathfrak {B}}}). Moreover, let ({({{mathfrak {A}}}_y,S_y):yin {Y}}) be a Q-disintegration of R (if ({{mathfrak {A}}}_y={{mathfrak {A}}}) for every (yin {Y}), then we have a regular conditional probability on ({{mathfrak {A}}}) with respect to Q) and let ({{mathfrak {C}}}) be a sub-(sigma )-algebra of ({{mathfrak {A}}}cap bigcap _{yin {Y}}{{mathfrak {A}}}_y). We prove that if (fin {{mathcal {L}}}^{infty }(R)) and ({{mathbb {E}}}_{{{mathfrak {C}}}otimes {{mathfrak {B}}}}(f)) is the conditional expectation of f with respect to ({{mathfrak {C}}}otimes {{mathfrak {B}}}), then for Q-almost every (yin {Y}) the y-section ([{{mathbb {E}}}_{{{mathfrak {C}}}otimes {{mathfrak {B}}}}(f)]^y) is a version of the conditional expectation of (f^y) with respect ({{mathfrak {C}}}) and (S_y). Moreover there exist a lifting (pi ) on ({{mathcal {L}}}^{infty }(widehat{R})) ((widehat{R}) is the completion of R) and liftings (sigma _y) on ({{mathcal {L}}}^{infty }(widehat{S_y})), (yin Y), such that
Both results are generalizations of Strauss et al. (Ann Prob 32:2389–2408, 2004), where ({{mathfrak {A}}}) was assumed to be separable in the Frechet-Nikodým pseudometric and of Macheras et al. (J Math Anal Appl 335:213–224, 2007), where R was assumed to be absolutely continuous with respect to the product measure (Potimes {Q}). Finally a characterization of stochastic processes possessing an equivalent measurable version is presented.
{"title":"Splitting of conditional expectations and liftings in product spaces","authors":"Kazimierz Musiał","doi":"10.1007/s11117-023-01027-y","DOIUrl":"https://doi.org/10.1007/s11117-023-01027-y","url":null,"abstract":"<p>Let <span>((X, {{mathfrak {A}}},P))</span> and <span>((Y, {{mathfrak {B}}},Q))</span> be two probability spaces and <i>R</i> be their skew product on the product <span>(sigma )</span>-algebra <span>({{mathfrak {A}}}otimes {{mathfrak {B}}})</span>. Moreover, let <span>({({{mathfrak {A}}}_y,S_y):yin {Y}})</span> be a <i>Q</i>-disintegration of <i>R</i> (if <span>({{mathfrak {A}}}_y={{mathfrak {A}}})</span> for every <span>(yin {Y})</span>, then we have a regular conditional probability on <span>({{mathfrak {A}}})</span> with respect to <i>Q</i>) and let <span>({{mathfrak {C}}})</span> be a sub-<span>(sigma )</span>-algebra of <span>({{mathfrak {A}}}cap bigcap _{yin {Y}}{{mathfrak {A}}}_y)</span>. We prove that if <span>(fin {{mathcal {L}}}^{infty }(R))</span> and <span>({{mathbb {E}}}_{{{mathfrak {C}}}otimes {{mathfrak {B}}}}(f))</span> is the conditional expectation of <i>f</i> with respect to <span>({{mathfrak {C}}}otimes {{mathfrak {B}}})</span>, then for <i>Q</i>-almost every <span>(yin {Y})</span> the <i>y</i>-section <span>([{{mathbb {E}}}_{{{mathfrak {C}}}otimes {{mathfrak {B}}}}(f)]^y)</span> is a version of the conditional expectation of <span>(f^y)</span> with respect <span>({{mathfrak {C}}})</span> and <span>(S_y)</span>. Moreover there exist a lifting <span>(pi )</span> on <span>({{mathcal {L}}}^{infty }(widehat{R}))</span> (<span>(widehat{R})</span> is the completion of <i>R</i>) and liftings <span>(sigma _y)</span> on <span>({{mathcal {L}}}^{infty }(widehat{S_y}))</span>, <span>(yin Y)</span>, such that </p><span>$$begin{aligned}{}[pi (f)]^y= sigma _yBigl ([pi (f)]^yBigr ) qquad hbox {for all} quad yin Yquad hbox {and}quad fin {{mathcal {L}}}^{infty }(widehat{R}). end{aligned}$$</span><p>Both results are generalizations of Strauss et al. (Ann Prob 32:2389–2408, 2004), where <span>({{mathfrak {A}}})</span> was assumed to be separable in the Frechet-Nikodým pseudometric and of Macheras et al. (J Math Anal Appl 335:213–224, 2007), where <i>R</i> was assumed to be absolutely continuous with respect to the product measure <span>(Potimes {Q})</span>. Finally a characterization of stochastic processes possessing an equivalent measurable version is presented.</p>","PeriodicalId":54596,"journal":{"name":"Positivity","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139585757","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 : 2024-01-19DOI: 10.1007/s11117-023-01024-1
Abstract
Every atomic JBW-algebra is known to be a direct sum of JBW-algebra factors of type I. Extending Kadison’s anti-lattice theorem, we show that each of these factors is a disjointness free anti-lattice. We characterise disjointness, bands, and disjointness preserving bijections with disjointness preserving inverses in direct sums of disjointness free anti-lattices and, therefore, in atomic JBW-algebras. We show that in unital JB-algebras the algebraic centre and the order theoretical centre are isomorphic. Moreover, the order theoretical centre is a Riesz space of multiplication operators. A survey of JBW-algebra factors of type I is included.
摘要 众所周知,每个原子 JBW 代数都是 JBW 代数 I 型因子的直和。通过扩展凯迪森反晶格定理,我们证明了这些因子中的每个因子都是无相交反晶格。我们描述了无相交反晶格的直和中的无相交、带和具有无相交保全反的无相交保全双射的特征,因此也描述了原子 JBW-代数中的无相交、带和具有无相交保全反的无相交保全双射的特征。我们证明,在单元 JB-数中,代数中心和阶论中心是同构的。此外,阶论中心是乘法算子的里兹空间。其中还包括对 I 型 JBW-algebra 因子的考察。
{"title":"Order theoretical structures in atomic JBW-algebras: disjointness, bands, and centres","authors":"","doi":"10.1007/s11117-023-01024-1","DOIUrl":"https://doi.org/10.1007/s11117-023-01024-1","url":null,"abstract":"<h3>Abstract</h3> <p>Every atomic JBW-algebra is known to be a direct sum of JBW-algebra factors of type I. Extending Kadison’s anti-lattice theorem, we show that each of these factors is a disjointness free anti-lattice. We characterise disjointness, bands, and disjointness preserving bijections with disjointness preserving inverses in direct sums of disjointness free anti-lattices and, therefore, in atomic JBW-algebras. We show that in unital JB-algebras the algebraic centre and the order theoretical centre are isomorphic. Moreover, the order theoretical centre is a Riesz space of multiplication operators. A survey of JBW-algebra factors of type I is included.</p>","PeriodicalId":54596,"journal":{"name":"Positivity","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139508765","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 : 2024-01-17DOI: 10.1007/s11117-023-01028-x
Michio Seto
In this paper, we give a new approach to the theory of kernel functions. Our method is based on the structure of Fock spaces. As its applications, two non-Euclidean examples of strictly positive kernel functions are given. Moreover, we give a new proof of the universal approximation theorem for Gaussian type kernels.
{"title":"A Fock space approach to the theory of kernel functions","authors":"Michio Seto","doi":"10.1007/s11117-023-01028-x","DOIUrl":"https://doi.org/10.1007/s11117-023-01028-x","url":null,"abstract":"<p>In this paper, we give a new approach to the theory of kernel functions. Our method is based on the structure of Fock spaces. As its applications, two non-Euclidean examples of strictly positive kernel functions are given. Moreover, we give a new proof of the universal approximation theorem for Gaussian type kernels.</p>","PeriodicalId":54596,"journal":{"name":"Positivity","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139483572","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 : 2024-01-12DOI: 10.1007/s11117-023-01026-z
Meenakshi Gupta, Manjari Srivastava
In this paper, we introduce two kinds of Hadamard well-posedness for a set optimization problem by taking into consideration perturbations of objective function and a relationship between these two well-posedness is derived. Using the generalized Gerstewitz function, a sequence of scalar optimization problems have been defined and a convergence result is obtained. Sufficient conditions for Hadamard well-posedness for the set optimization problem are established by invoking these scalarization results. Finally, the stability of the convergence of minimal solution sets of the set optimization problem considered is discussed in terms of Painlevé-Kuratowski convergence.
{"title":"Hadamard well-posedness and stability in set optimization","authors":"Meenakshi Gupta, Manjari Srivastava","doi":"10.1007/s11117-023-01026-z","DOIUrl":"https://doi.org/10.1007/s11117-023-01026-z","url":null,"abstract":"<p>In this paper, we introduce two kinds of Hadamard well-posedness for a set optimization problem by taking into consideration perturbations of objective function and a relationship between these two well-posedness is derived. Using the generalized Gerstewitz function, a sequence of scalar optimization problems have been defined and a convergence result is obtained. Sufficient conditions for Hadamard well-posedness for the set optimization problem are established by invoking these scalarization results. Finally, the stability of the convergence of minimal solution sets of the set optimization problem considered is discussed in terms of Painlevé-Kuratowski convergence.\u0000</p>","PeriodicalId":54596,"journal":{"name":"Positivity","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139461444","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 : 2024-01-12DOI: 10.1007/s11117-023-01025-0
Volodymyr Mykhaylyuk, Marat Pliev, Mikhail Popov
The present paper aims to describe the relationships between the intersection property, introduced and studied in the previous paper by the authors, with other known properties of Riesz spaces, and to prove that every lateral ideal of a Riesz space is a kernel of some positive orthogonally additive operator (it is easy to see that the kernel of every positive orthogonally additive operator is a lateral ideal). We provide examples of Riesz spaces with the principal projection property (and hence, with the intersection property) which fail to be C-complete. The above results give complete answers to problems posed in the first part of the present paper by the authors.
{"title":"The lateral order on Riesz spaces and orthogonally additive operators. II","authors":"Volodymyr Mykhaylyuk, Marat Pliev, Mikhail Popov","doi":"10.1007/s11117-023-01025-0","DOIUrl":"https://doi.org/10.1007/s11117-023-01025-0","url":null,"abstract":"<p>The present paper aims to describe the relationships between the intersection property, introduced and studied in the previous paper by the authors, with other known properties of Riesz spaces, and to prove that every lateral ideal of a Riesz space is a kernel of some positive orthogonally additive operator (it is easy to see that the kernel of every positive orthogonally additive operator is a lateral ideal). We provide examples of Riesz spaces with the principal projection property (and hence, with the intersection property) which fail to be <i>C</i>-complete. The above results give complete answers to problems posed in the first part of the present paper by the authors.</p>","PeriodicalId":54596,"journal":{"name":"Positivity","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139461622","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}