Metaplectic operators form a relevant class of operators appearing in different applications, in this work we study their Schwartz kernels. Namely, diagonality of a kernel is defined by imposing rapid off-diagonal decay conditions, and quasi-diagonality by imposing the same conditions on the smoothing of the kernel through convolution with the Gaussian. Kernels of metaplectic operators are not diagonal. Nevertheless, as we shall prove, they are quasi-diagonal under suitable conditions. Motivation for our study comes from problems in time–frequency analysis, that we discuss in the last section.
{"title":"Metaplectic operators with quasi-diagonal kernels","authors":"Gianluca Giacchi, Luigi Rodino","doi":"10.1002/mana.70053","DOIUrl":"https://doi.org/10.1002/mana.70053","url":null,"abstract":"<p>Metaplectic operators form a relevant class of operators appearing in different applications, in this work we study their Schwartz kernels. Namely, diagonality of a kernel is defined by imposing rapid off-diagonal decay conditions, and quasi-diagonality by imposing the same conditions on the smoothing of the kernel through convolution with the Gaussian. Kernels of metaplectic operators are not diagonal. Nevertheless, as we shall prove, they are quasi-diagonal under suitable conditions. Motivation for our study comes from problems in time–frequency analysis, that we discuss in the last section.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 12","pages":"3618-3638"},"PeriodicalIF":0.8,"publicationDate":"2025-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.70053","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145719802","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p>Let <span></span><math> <semantics> <mi>S</mi> <annotation>$S$</annotation> </semantics></math> be a subset of <span></span><math> <semantics> <mover> <mi>Z</mi> <mo>¯</mo> </mover> <annotation>$overline{mathbb {Z}}$</annotation> </semantics></math>, the ring of all algebraic integers. A polynomial <span></span><math> <semantics> <mrow> <mi>f</mi> <mo>∈</mo> <mi>Q</mi> <mo>[</mo> <mi>X</mi> <mo>]</mo> </mrow> <annotation>$f in mathbb {Q}[X]$</annotation> </semantics></math> is said to be integral-valued on <span></span><math> <semantics> <mi>S</mi> <annotation>$S$</annotation> </semantics></math> if <span></span><math> <semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>s</mi> <mo>)</mo> </mrow> <mo>∈</mo> <mover> <mi>Z</mi> <mo>¯</mo> </mover> </mrow> <annotation>$f(s) in overline{mathbb {Z}}$</annotation> </semantics></math> for all <span></span><math> <semantics> <mrow> <mi>s</mi> <mo>∈</mo> <mi>S</mi> </mrow> <annotation>$s in S$</annotation> </semantics></math>. The set <span></span><math> <semantics> <mrow> <msub> <mi>Int</mi> <mi>Q</mi> </msub> <mrow> <mo>(</mo> <mi>S</mi> <mo>,</mo> <mover> <mi>Z</mi> <mo>¯</mo> </mover> <mo>)</mo> </mrow> </mrow> <annotation>${mathrm{Int}}_{mathbb{Q}}(S,bar{mathbb{Z}})$</annotation> </semantics></math> of all integral-valued polynomials on <span></span><math> <semantics> <mi>S</mi> <annotation>$S$</annotation> </semantics></math> forms a subring of <span></span><math> <semantics> <mrow> <mi>Q</mi> <mo>[</mo> <
{"title":"Nontriviality of rings of integral-valued polynomials","authors":"Giulio Peruginelli, Nicholas J. Werner","doi":"10.1002/mana.70057","DOIUrl":"https://doi.org/10.1002/mana.70057","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math> be a subset of <span></span><math>\u0000 <semantics>\u0000 <mover>\u0000 <mi>Z</mi>\u0000 <mo>¯</mo>\u0000 </mover>\u0000 <annotation>$overline{mathbb {Z}}$</annotation>\u0000 </semantics></math>, the ring of all algebraic integers. A polynomial <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mo>∈</mo>\u0000 <mi>Q</mi>\u0000 <mo>[</mo>\u0000 <mi>X</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <annotation>$f in mathbb {Q}[X]$</annotation>\u0000 </semantics></math> is said to be integral-valued on <span></span><math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math> if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>s</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>∈</mo>\u0000 <mover>\u0000 <mi>Z</mi>\u0000 <mo>¯</mo>\u0000 </mover>\u0000 </mrow>\u0000 <annotation>$f(s) in overline{mathbb {Z}}$</annotation>\u0000 </semantics></math> for all <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>s</mi>\u0000 <mo>∈</mo>\u0000 <mi>S</mi>\u0000 </mrow>\u0000 <annotation>$s in S$</annotation>\u0000 </semantics></math>. The set <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>Int</mi>\u0000 <mi>Q</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>S</mi>\u0000 <mo>,</mo>\u0000 <mover>\u0000 <mi>Z</mi>\u0000 <mo>¯</mo>\u0000 </mover>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>${mathrm{Int}}_{mathbb{Q}}(S,bar{mathbb{Z}})$</annotation>\u0000 </semantics></math> of all integral-valued polynomials on <span></span><math>\u0000 <semantics>\u0000 <mi>S</mi>\u0000 <annotation>$S$</annotation>\u0000 </semantics></math> forms a subring of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Q</mi>\u0000 <mo>[</mo>\u0000 <","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 12","pages":"3974-3994"},"PeriodicalIF":0.8,"publicationDate":"2025-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.70057","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145719833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove a generalized version of the Principle for Green's functions on bounded inner uniform domains in a wide class of Dirichlet spaces. In particular, our results apply to higher-dimensional fractals such as Sierpinski carpets in , , as well as generalized fractal-type spaces that do not have a well-defined Hausdorff dimension or walk dimension. This yields new instances of the principle for these spaces. We also discuss applications to Schrödinger operators.
{"title":"The boundary Harnack principle and the 3G principle in fractal-type spaces","authors":"Anthony Graves-McCleary, Laurent Saloff-Coste","doi":"10.1002/mana.70059","DOIUrl":"https://doi.org/10.1002/mana.70059","url":null,"abstract":"<p>We prove a generalized version of the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation>$3G$</annotation>\u0000 </semantics></math> Principle for Green's functions on bounded inner uniform domains in a wide class of Dirichlet spaces. In particular, our results apply to higher-dimensional fractals such as Sierpinski carpets in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>$mathbf {R}^n$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>≥</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$nge 3$</annotation>\u0000 </semantics></math>, as well as generalized fractal-type spaces that do not have a well-defined Hausdorff dimension or walk dimension. This yields new instances of the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation>$3G$</annotation>\u0000 </semantics></math> principle for these spaces. We also discuss applications to Schrödinger operators.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 11","pages":"3554-3575"},"PeriodicalIF":0.8,"publicationDate":"2025-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145486972","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 develop a method of evaluating general exponential sums with rational amplitude functions for multiple variables, which complements work by T. Cochrane and Z. Zheng on the single variable case. As an application, for , a fixed natural number, we obtain an asymptotic formula for the (weighted) number of solutions of general quadratic congruences modulo in small boxes, thus establishing an equidistribution result for these solutions.
{"title":"Multiple exponential sums and their applications to quadratic congruences","authors":"Nilanjan Bag, Stephan Baier, Anup Haldar","doi":"10.1002/mana.70063","DOIUrl":"https://doi.org/10.1002/mana.70063","url":null,"abstract":"<p>In this paper, we develop a method of evaluating general exponential sums with rational amplitude functions for multiple variables, which complements work by T. Cochrane and Z. Zheng on the single variable case. As an application, for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>≥</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$nge 3$</annotation>\u0000 </semantics></math>, a fixed natural number, we obtain an asymptotic formula for the (weighted) number of solutions of general quadratic congruences modulo <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>p</mi>\u0000 <mi>m</mi>\u0000 </msup>\u0000 <annotation>$p^m$</annotation>\u0000 </semantics></math> in small boxes, thus establishing an equidistribution result for these solutions.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 11","pages":"3599-3612"},"PeriodicalIF":0.8,"publicationDate":"2025-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145486970","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 deals with the qualitative properties of solutions to the Cauchy problem for a quasilinear convection–diffusion equation with volumetric moisture content, which involves the cases of critical and fast decaying volumetric moisture content. Based on the method of directly constructing Barenblatt-type super- and sub-solutions, we establish the new criteria of global existence and nonexistence and provide the properties of expansion and shrinking of the support of the weak solution.
{"title":"Global existence and nonexistence for an inhomogeneous quasilinear convection-diffusion equation","authors":"Wentao Huo, Zhong Bo Fang","doi":"10.1002/mana.70056","DOIUrl":"https://doi.org/10.1002/mana.70056","url":null,"abstract":"<p>This paper deals with the qualitative properties of solutions to the Cauchy problem for a quasilinear convection–diffusion equation with volumetric moisture content, which involves the cases of critical and fast decaying volumetric moisture content. Based on the method of directly constructing Barenblatt-type super- and sub-solutions, we establish the new criteria of global existence and nonexistence and provide the properties of expansion and shrinking of the support of the weak solution.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 11","pages":"3515-3539"},"PeriodicalIF":0.8,"publicationDate":"2025-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145486971","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}
D. L. Fernandez, M. Mastyło, J. Santos, E. B. Silva
<p>We study lattice summing operators between Banach spaces focusing on two classes, <span></span><math> <semantics> <msub> <mi>ℓ</mi> <mi>φ</mi> </msub> <annotation>$ell _varphi$</annotation> </semantics></math>-summing and strongly <span></span><math> <semantics> <mi>φ</mi> <annotation>$varphi$</annotation> </semantics></math>-summing operators, which are generated by Orlicz sequence lattices <span></span><math> <semantics> <msub> <mi>ℓ</mi> <mi>φ</mi> </msub> <annotation>$ell _varphi$</annotation> </semantics></math>. For the class of strongly <span></span><math> <semantics> <mi>φ</mi> <annotation>$varphi$</annotation> </semantics></math>-summing operators, we prove the domination theorem, which complements Pietsch's fundamental domination theorem for <span></span><math> <semantics> <mi>p</mi> <annotation>$p$</annotation> </semantics></math>-summing operators. Based on this result, we show that strongly <span></span><math> <semantics> <mi>φ</mi> <annotation>$varphi$</annotation> </semantics></math>-summing operators are Dunford–Pettis. As a consequence, we show that these classes are, in general, distinct. We also demonstrate that the class of strongly <span></span><math> <semantics> <mi>φ</mi> <annotation>$varphi$</annotation> </semantics></math>-summing operators between Hilbert spaces coincides with the Hilbert–Schmidt class when <span></span><math> <semantics> <msub> <mi>ℓ</mi> <mi>φ</mi> </msub> <annotation>$ell _varphi$</annotation> </semantics></math> is a separable Orlicz space. Finally, we consider generalized nuclear operators, and using a factorization description, we prove that <span></span><math> <semantics> <msub> <mi>ℓ</mi> <mi>φ</mi> </msub> <annotation>$ell _varphi$</annotation> </semantics></math>-nuclear operators are <span></span><math> <semantics> <msub> <mi>ℓ</mi> <mi>φ</mi> </msub> <annotation>$ell _varphi$</annotation> </semantics></math>-summing when <span></span><math> <semantics> <msub> <mi>ℓ</mi> <mi>
{"title":"The domination theorem for operator classes generated by Orlicz spaces","authors":"D. L. Fernandez, M. Mastyło, J. Santos, E. B. Silva","doi":"10.1002/mana.70060","DOIUrl":"https://doi.org/10.1002/mana.70060","url":null,"abstract":"<p>We study lattice summing operators between Banach spaces focusing on two classes, <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℓ</mi>\u0000 <mi>φ</mi>\u0000 </msub>\u0000 <annotation>$ell _varphi$</annotation>\u0000 </semantics></math>-summing and strongly <span></span><math>\u0000 <semantics>\u0000 <mi>φ</mi>\u0000 <annotation>$varphi$</annotation>\u0000 </semantics></math>-summing operators, which are generated by Orlicz sequence lattices <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℓ</mi>\u0000 <mi>φ</mi>\u0000 </msub>\u0000 <annotation>$ell _varphi$</annotation>\u0000 </semantics></math>. For the class of strongly <span></span><math>\u0000 <semantics>\u0000 <mi>φ</mi>\u0000 <annotation>$varphi$</annotation>\u0000 </semantics></math>-summing operators, we prove the domination theorem, which complements Pietsch's fundamental domination theorem for <span></span><math>\u0000 <semantics>\u0000 <mi>p</mi>\u0000 <annotation>$p$</annotation>\u0000 </semantics></math>-summing operators. Based on this result, we show that strongly <span></span><math>\u0000 <semantics>\u0000 <mi>φ</mi>\u0000 <annotation>$varphi$</annotation>\u0000 </semantics></math>-summing operators are Dunford–Pettis. As a consequence, we show that these classes are, in general, distinct. We also demonstrate that the class of strongly <span></span><math>\u0000 <semantics>\u0000 <mi>φ</mi>\u0000 <annotation>$varphi$</annotation>\u0000 </semantics></math>-summing operators between Hilbert spaces coincides with the Hilbert–Schmidt class when <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℓ</mi>\u0000 <mi>φ</mi>\u0000 </msub>\u0000 <annotation>$ell _varphi$</annotation>\u0000 </semantics></math> is a separable Orlicz space. Finally, we consider generalized nuclear operators, and using a factorization description, we prove that <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℓ</mi>\u0000 <mi>φ</mi>\u0000 </msub>\u0000 <annotation>$ell _varphi$</annotation>\u0000 </semantics></math>-nuclear operators are <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℓ</mi>\u0000 <mi>φ</mi>\u0000 </msub>\u0000 <annotation>$ell _varphi$</annotation>\u0000 </semantics></math>-summing when <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>ℓ</mi>\u0000 <mi>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 11","pages":"3576-3598"},"PeriodicalIF":0.8,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.70060","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145486818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider the following Schrödinger–Poisson–Slater equation:
本文考虑如下Schrödinger-Poisson-Slater方程：
{"title":"Ground states for a zero-mass and Coulomb–Sobolev critical Schrödinger–Poisson–Slater problem","authors":"Xiaoquan Feng, Xingwen Chen, Qiongfen Zhang","doi":"10.1002/mana.70058","DOIUrl":"https://doi.org/10.1002/mana.70058","url":null,"abstract":"<p>In this paper, we consider the following Schrödinger–Poisson–Slater equation:\u0000\u0000 </p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 11","pages":"3540-3553"},"PeriodicalIF":0.8,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145486956","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 study the dynamics of the focusing nonlinear Hartree equation with a Kato potential
本文研究了具有加藤势的聚焦非线性Hartree方程的动力学问题
{"title":"The dynamics of the focusing NLH with a potential beyond the mass–energy threshold","authors":"Shuang Ji, Jing Lu, Fanfei Meng","doi":"10.1002/mana.70047","DOIUrl":"https://doi.org/10.1002/mana.70047","url":null,"abstract":"<p>In this paper, we study the dynamics of the focusing nonlinear Hartree equation with a Kato potential\u0000\u0000 </p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 11","pages":"3444-3459"},"PeriodicalIF":0.8,"publicationDate":"2025-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145486676","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}
Assume that the section conjecture holds over number fields. We prove then that it holds for a broad class of curves defined over finitely generated extensions of . This class contains every projective, hyperelliptic curve, every hyperbolic, affine curve of genus , and a basis of open subsets of any curve. If we furthermore assume the weak Bombieri–Lang conjecture, we prove that the section conjecture holds for every hyperbolic curve over every finitely generated extension of .
{"title":"On the section conjecture over fields of finite type","authors":"Giulio Bresciani","doi":"10.1002/mana.70049","DOIUrl":"https://doi.org/10.1002/mana.70049","url":null,"abstract":"<p>Assume that the section conjecture holds over number fields. We prove then that it holds for a broad class of curves defined over finitely generated extensions of <span></span><math>\u0000 <semantics>\u0000 <mi>Q</mi>\u0000 <annotation>$mathbb {Q}$</annotation>\u0000 </semantics></math>. This class contains every projective, hyperelliptic curve, every hyperbolic, affine curve of genus <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>≤</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$le 2$</annotation>\u0000 </semantics></math>, and a basis of open subsets of any curve. If we furthermore assume the weak Bombieri–Lang conjecture, we prove that the section conjecture holds for every hyperbolic curve over every finitely generated extension of <span></span><math>\u0000 <semantics>\u0000 <mi>Q</mi>\u0000 <annotation>$mathbb {Q}$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49853,"journal":{"name":"Mathematische Nachrichten","volume":"298 11","pages":"3476-3493"},"PeriodicalIF":0.8,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mana.70049","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145486692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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