Mathematika最新文献

{"title":"A dichotomy phenomenon for bad minus normed Dirichlet","authors":"Dmitry Kleinbock,&nbsp;Anurag Rao","doi":"10.1112/mtk.12221","DOIUrl":"10.1112/mtk.12221","url":null,"abstract":"<p>Given a norm ν on <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <annotation>$mathbb {R}^2$</annotation>\u0000 </semantics></math>, the set of ν-Dirichlet improvable numbers <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>DI</mi>\u0000 <mi>ν</mi>\u0000 </msub>\u0000 <annotation>$mathbf {DI}_nu$</annotation>\u0000 </semantics></math> was defined and studied in the papers (Andersen and Duke, <i>Acta Arith</i>. 198 (2021) 37–75 and Kleinbock and Rao, <i>Internat. Math. Res. Notices</i> 2022 (2022) 5617–5657). When ν is the supremum norm, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>DI</mi>\u0000 <mi>ν</mi>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <mi>BA</mi>\u0000 <mo>∪</mo>\u0000 <mi>Q</mi>\u0000 </mrow>\u0000 <annotation>$mathbf {DI}_nu = mathbf {BA}cup {mathbb {Q}}$</annotation>\u0000 </semantics></math>, where <math>\u0000 <semantics>\u0000 <mi>BA</mi>\u0000 <annotation>$mathbf {BA}$</annotation>\u0000 </semantics></math> is the set of badly approximable numbers. Each of the sets <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>DI</mi>\u0000 <mi>ν</mi>\u0000 </msub>\u0000 <annotation>$mathbf {DI}_nu$</annotation>\u0000 </semantics></math>, like <math>\u0000 <semantics>\u0000 <mi>BA</mi>\u0000 <annotation>$mathbf {BA}$</annotation>\u0000 </semantics></math>, is of measure zero and satisfies the winning property of Schmidt. Hence for every norm ν, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>BA</mi>\u0000 <mo>∩</mo>\u0000 <msub>\u0000 <mi>DI</mi>\u0000 <mi>ν</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$mathbf {BA} cap mathbf {DI}_nu$</annotation>\u0000 </semantics></math> is winning and thus has full Hausdorff dimension. In this article, we prove the following dichotomy phenomenon: either <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>BA</mi>\u0000 <mo>⊂</mo>\u0000 <msub>\u0000 <mi>DI</mi>\u0000 <mi>ν</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$mathbf {BA} subset mathbf {DI}_nu$</annotation>\u0000 </semantics></math> or else <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>BA</mi>\u0000 <mo>∖</mo>\u0000 <msub>\u0000 <mi>DI</mi>\u0000 <mi>ν</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$mathbf {BA} setminus mathbf {DI}_nu$</annotation>\u0000 ","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46726104","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}

{"title":"Families of ϕ-congruence subgroups of the modular group","authors":"Angelica Babei,&nbsp;Andrew Fiori,&nbsp;Cameron Franc","doi":"10.1112/mtk.12218","DOIUrl":"10.1112/mtk.12218","url":null,"abstract":"We introduce and study families of finite index subgroups of the modular group that generalize the congruence subgroups. Such groups, termed ϕ‐congruence subgroups, are obtained by reducing homomorphisms ϕ from the modular group into a linear algebraic group modulo integers. In particular, we examine two families of examples, arising on the one hand from a map into a quasi‐unipotent group, and on the other hand from maps into symplectic groups of degree four. In the quasi‐unipotent case, we also provide a detailed discussion of the corresponding modular forms, using the fact that the tower of curves in this case contains the tower of isogenies over the elliptic curve y2=x3−1728$y^2=x^3-1728$ defined by the commutator subgroup of the modular group.","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12218","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45504633","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}

{"title":"Lower bounds for negative moments of \u0000 \u0000 \u0000 \u0000 ζ\u0000 ′\u0000 \u0000 \u0000 (\u0000 ρ\u0000 )\u0000 \u0000 \u0000 $zeta ^{prime }(rho )$","authors":"Peng Gao,&nbsp;Liangyi Zhao","doi":"10.1112/mtk.12219","DOIUrl":"10.1112/mtk.12219","url":null,"abstract":"<p>We establish lower bounds for the discrete 2<i>k</i>th moment of the derivative of the Riemann zeta function at nontrivial zeros for all <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mo>&lt;</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$k&lt;0$</annotation>\u0000 </semantics></math> under the Riemann hypothesis and the assumption that all zeros of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ζ</mi>\u0000 <mo>(</mo>\u0000 <mi>s</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$zeta (s)$</annotation>\u0000 </semantics></math> are simple.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12219","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42508800","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}

{"title":"Averages of long Dirichlet polynomials with modular coefficients","authors":"Brian Conrey,&nbsp;Alessandro Fazzari","doi":"10.1112/mtk.12220","DOIUrl":"10.1112/mtk.12220","url":null,"abstract":"<p>We study the moments of <i>L</i>-functions associated with primitive cusp forms, in the weight aspect. In particular, we obtain an asymptotic formula for the twisted moments of a <i>long</i> Dirichlet polynomial with modular coefficients. This result, which is conditional on the Generalized Lindelöf Hypothesis, agrees with the prediction of the recipe by Conrey, Farmer, Keating, Rubinstein and Snaith.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46217938","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}

{"title":"M0, 5: Toward the Chabauty–Kim method in higher dimensions","authors":"Ishai Dan-Cohen,&nbsp;David Jarossay","doi":"10.1112/mtk.12215","DOIUrl":"10.1112/mtk.12215","url":null,"abstract":"<p>If <i>Z</i> is an open subscheme of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>Spec</mo>\u0000 <mi>Z</mi>\u0000 </mrow>\u0000 <annotation>$operatorname{Spec}mathbb {Z}$</annotation>\u0000 </semantics></math>, <i>X</i> is a sufficiently nice <i>Z</i>-model of a smooth curve over <math>\u0000 <semantics>\u0000 <mi>Q</mi>\u0000 <annotation>$mathbb {Q}$</annotation>\u0000 </semantics></math>, and <i>p</i> is a closed point of <i>Z</i>, the Chabauty–Kim method leads to the construction of locally analytic functions on <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 <mi>p</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$X({mathbb {Z}_p})$</annotation>\u0000 </semantics></math> which vanish on <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 <mo>(</mo>\u0000 <mi>Z</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$X(Z)$</annotation>\u0000 </semantics></math>; we call such functions “Kim functions”. At least in broad outline, the method generalizes readily to higher dimensions. In fact, in some sense, the surface <i>M</i>_{0, 5} should be easier than the previously studied curve <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>M</mi>\u0000 <mrow>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mn>4</mn>\u0000 </mrow>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <mo>∖</mo>\u0000 <mrow>\u0000 <mo>{</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mi>∞</mi>\u0000 <mo>}</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$M_{0,4} = mathbb {P}^1 setminus lbrace 0,1,infty rbrace$</annotation>\u0000 </semantics></math> since its points are closely related to those of <i>M</i>_{0, 4}, yet they face a further condition to integrality. This is mirrored by a certain <i>weight advantage</i> we encounter, because of which, <i>M</i>_{0, 5} possesses <i>new Kim functions</i> not coming from <i>M</i>_{0, 4}. Here we focus on the case “<math>\u0000 <semantics>\u0000 <mrow>\u0000 ","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12215","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41296649","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}

{"title":"Banach spaces of continuous functions without norming Markushevich bases","authors":"Tommaso Russo,&nbsp;Jacopo Somaglia","doi":"10.1112/mtk.12217","DOIUrl":"10.1112/mtk.12217","url":null,"abstract":"<p>We investigate the question whether a scattered compact topological space <i>K</i> such that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <mo>(</mo>\u0000 <mi>K</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$C(K)$</annotation>\u0000 </semantics></math> has a norming Markushevich basis (M-basis, for short) must be Eberlein. This question originates from the recent solution, due to Hájek, Todorčević and the authors, to an open problem from the 1990s, due to Godefroy. Our prime tool consists in proving that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$C([0,omega _1])$</annotation>\u0000 </semantics></math> does not embed in a Banach space with a norming M-basis, thereby generalising a result due to Alexandrov and Plichko. Subsequently, we give sufficient conditions on a compact <i>K</i> for <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 <mo>(</mo>\u0000 <mi>K</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$C(K)$</annotation>\u0000 </semantics></math> not to embed in a Banach space with a norming M-basis. Examples of such conditions are that <i>K</i> is a zero-dimensional compact space with a P-point, or a compact tree of height at least <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$omega _1 +1$</annotation>\u0000 </semantics></math>. In particular, this allows us to answer the said question in the case when <i>K</i> is a tree and to obtain a rather general result for Valdivia compacta. Finally, we give some structural results for scattered compact trees; in particular, we prove that scattered trees of height less than ω₂ are Valdivia.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12217","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47910696","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}

{"title":"On simply normal numbers with digit dependencies","authors":"Verónica Becher,&nbsp;Agustín Marchionna,&nbsp;Gérald Tenenbaum","doi":"10.1112/mtk.12216","DOIUrl":"10.1112/mtk.12216","url":null,"abstract":"<p>Given an integer <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>b</mi>\u0000 <mo>⩾</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$bgeqslant 2$</annotation>\u0000 </semantics></math> and a set <math>\u0000 <semantics>\u0000 <mi>P</mi>\u0000 <annotation>${EuScript P}$</annotation>\u0000 </semantics></math> of prime numbers, the set <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>T</mi>\u0000 <mi>P</mi>\u0000 </msub>\u0000 <annotation>${EuScript T}_{EuScript P}$</annotation>\u0000 </semantics></math> of Toeplitz numbers comprises all elements of [0, <i>b</i>[ whose digits <math>\u0000 <semantics>\u0000 <msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>a</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>⩾</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$(a_n)_{ngeqslant 1}$</annotation>\u0000 </semantics></math> in the base-<i>b</i> expansion satisfy <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>a</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <msub>\u0000 <mi>a</mi>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$a_n=a_{pn}$</annotation>\u0000 </semantics></math> for all <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>∈</mo>\u0000 <mi>P</mi>\u0000 </mrow>\u0000 <annotation>$pin {EuScript P}$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>⩾</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$ngeqslant 1$</annotation>\u0000 </semantics></math>. Using a completely additive arithmetical function, we construct a number in <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>T</mi>\u0000 <mi>P</mi>\u0000 </msub>\u0000 <annotation>${EuScript T}_{EuScript P}$</annotation>\u0000 </semantics></math> that is simply Borel normal if, and only if, <math>\u0000 <semantics>\u0000 <mstyle>\u0000 <mrow>\u0000 <msub>\u0000 <mo>∑</mo>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>∉</mo>\u0000 ","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46557821","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}

{"title":"Rational points close to non-singular algebraic curves","authors":"Faustin Adiceam,&nbsp;Oscar Marmon","doi":"10.1112/mtk.12214","DOIUrl":"10.1112/mtk.12214","url":null,"abstract":"<p>We study the density of solutions to Diophantine inequalities involving non-singular ternary forms, or equivalently, the density of rational points close to non-singular plane algebraic curves.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12214","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49630476","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}

{"title":"The functional Orlicz–Brunn–Minkowski inequality for q-torsional rigidity","authors":"Jinrong Hu,&nbsp;Ping Zhang","doi":"10.1112/mtk.12213","DOIUrl":"10.1112/mtk.12213","url":null,"abstract":"<p>In this paper, we obtain the functional Orlicz–Brunn–Minkowski inequality and the functional Orlicz–Minkowski inequality for <i>q</i>-torsional rigidity in the smooth category. Furthermore, using an approximation method, we give the general functional Orlicz–Brunn–Minkowski inequality for <i>q</i>-torsional rigidity. As a corollary, we reveal that the functional Orlicz–Brunn–Minkowski inequality is equivalent to the functional Orlicz–Minkowski inequality for <i>q</i>-torsional rigidity in the smooth category. We also give some applications with respect to these two inequalities.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45701642","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}

{"title":"On the distribution of equivalence classes of random symmetric p-adic matrices","authors":"Valeriya Kovaleva","doi":"10.1112/mtk.12212","DOIUrl":"https://doi.org/10.1112/mtk.12212","url":null,"abstract":"<p>We consider random symmetric matrices with independent entries distributed according to the Haar measure on <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 <mi>p</mi>\u0000 </msub>\u0000 <annotation>$mathbb {Z}_p$</annotation>\u0000 </semantics></math> for odd primes <i>p</i> and derive the distribution of their canonical form with respect to several equivalence relations. We give a few examples of applications including an alternative proof for the result of Bhargava, Cremona, Fisher, Jones and Keating on the probability that a random quadratic form over <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 <mi>p</mi>\u0000 </msub>\u0000 <annotation>$mathbb {Z}_p$</annotation>\u0000 </semantics></math> has a non-trivial zero.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12212","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50152399","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}