Mathematika最新文献

{"title":"On the \u0000 \u0000 \u0000 a\u0000 b\u0000 c\u0000 \u0000 $abc$\u0000 conjecture in algebraic number fields","authors":"Andrew Scoones","doi":"10.1112/mtk.12230","DOIUrl":"https://doi.org/10.1112/mtk.12230","url":null,"abstract":"<p>In this paper, we prove a weak form of the <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mi>b</mi>\u0000 <mi>c</mi>\u0000 </mrow>\u0000 <annotation>$abc$</annotation>\u0000 </semantics></math> conjecture generalised to algebraic number fields. Given integers satisfying <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mo>+</mo>\u0000 <mi>b</mi>\u0000 <mo>=</mo>\u0000 <mi>c</mi>\u0000 </mrow>\u0000 <annotation>$a+b=c$</annotation>\u0000 </semantics></math>, Stewart and Yu were able to give an exponential bound in terms of the radical over the integers (Stewart and Yu [Math. Ann. <b>291</b> (1991), 225–230], Stewart and Yu [Duke Math. J. <b>108</b> (2001), no. 1, 169–181]), whereas Győry was able to give an exponential bound in the algebraic number field case for the projective height <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>H</mi>\u0000 <mi>K</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>a</mi>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <mi>b</mi>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <mi>c</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$H_{K}(a,,b,,c)$</annotation>\u0000 </semantics></math> in terms of the radical for algebraic numbers (Győry [Acta Arith. <b>133</b> (2008), 281–295]). We generalise Stewart and Yu's method to give an improvement on Győry's bound for algebraic integers over the Hilbert Class Field of the initial number field <i>K</i>. Given algebraic integers <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <mi>b</mi>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <mi>c</mi>\u0000 </mrow>\u0000 <annotation>$a,,b,,c$</annotation>\u0000 </semantics></math> in a number field <i>K</i> satisfying <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mo>+</mo>\u0000 <mi>b</mi>\u0000 <mo>=</mo>\u0000 <mi>c</mi>\u0000 </mrow>\u0000 <annotation>$a+b=c$</annotation>\u0000 </semantics></math>, we give an upper bound for the logarithm of the projective height <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>H</","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"70 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12230","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68181298","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 the number of tiles visited by a line segment on a rectangular grid","authors":"Alex Arkhipov,&nbsp;Luis Mendo","doi":"10.1112/mtk.12223","DOIUrl":"https://doi.org/10.1112/mtk.12223","url":null,"abstract":"<p>Consider a line segment placed on a two-dimensional grid of rectangular tiles. This paper addresses the relationship between the length of the segment and the number of tiles it visits (i.e., has intersection with). The square grid is also considered explicitly, as some of the specific problems studied are more tractable in that particular case. The segment position and orientation can be modeled as either deterministic or random. In the deterministic setting, the maximum possible number of visited tiles is characterized for a given length, and conversely, the infimum segment length needed to visit a desired number of tiles is analyzed. In the random setting, the average number of visited tiles and the probability of visiting the maximum number of tiles on a square grid are studied as a function of segment length. These questions are related to Buffon's needle problem and its extension by Laplace.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 4","pages":"1242-1281"},"PeriodicalIF":0.8,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12223","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50149073","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":"Optimal Hardy-weights for elliptic operators with mixed boundary conditions","authors":"Yehuda Pinchover,&nbsp;Idan Versano","doi":"10.1112/mtk.12226","DOIUrl":"https://doi.org/10.1112/mtk.12226","url":null,"abstract":"<p>We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>P</mi>\u0000 <mo>,</mo>\u0000 <mi>B</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(P,B)$</annotation>\u0000 </semantics></math> with degenerate mixed boundary conditions. By an optimal Hardy-weight for a subcritical operator we mean a nonzero nonnegative weight function <i>W</i> such that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>P</mi>\u0000 <mo>−</mo>\u0000 <mi>W</mi>\u0000 <mo>,</mo>\u0000 <mi>B</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(P-W,B)$</annotation>\u0000 </semantics></math> is critical, and null-critical with respect to <i>W</i>. Our results rely on a recently developed criticality theory for positive solutions of the corresponding mixed boundary value problem.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 4","pages":"1221-1241"},"PeriodicalIF":0.8,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12226","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50147213","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":"Stability of polydisc slicing","authors":"Nathaniel Glover,&nbsp;Tomasz Tkocz,&nbsp;Katarzyna Wyczesany","doi":"10.1112/mtk.12225","DOIUrl":"https://doi.org/10.1112/mtk.12225","url":null,"abstract":"<p>We prove a dimension-free stability result for polydisc slicing due to Oleszkiewicz and Pełczyński. Intriguingly, compared to the real case, there is an additional asymptotic maximizer. In addition to Fourier-analytic bounds, we crucially rely on a self-improving feature of polydisc slicing, established via probabilistic arguments.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 4","pages":"1165-1182"},"PeriodicalIF":0.8,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12225","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50145819","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":"Two problems on the distribution of Carmichael's lambda function","authors":"Paul Pollack","doi":"10.1112/mtk.12222","DOIUrl":"https://doi.org/10.1112/mtk.12222","url":null,"abstract":"<p>Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$lambda (n)$</annotation>\u0000 </semantics></math> denote the exponent of the multiplicative group modulo <i>n</i>. We show that when <i>q</i> is odd, each coprime residue class modulo <i>q</i> is hit equally often by <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$lambda (n)$</annotation>\u0000 </semantics></math> as <i>n</i> varies. Under the stronger assumption that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>gcd</mo>\u0000 <mo>(</mo>\u0000 <mi>q</mi>\u0000 <mo>,</mo>\u0000 <mn>6</mn>\u0000 <mo>)</mo>\u0000 <mo>=</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$gcd (q,6)=1$</annotation>\u0000 </semantics></math>, we prove that equidistribution persists throughout a Siegel–Walfisz-type range of uniformity. By similar methods we show that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$lambda (n)$</annotation>\u0000 </semantics></math> obeys Benford's leading digit law with respect to natural density. Moreover, if we assume Generalized Riemann Hypothesis, then Benford's law holds for the order of <i>a</i> mod <i>n</i>, for any fixed integer <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mo>∉</mo>\u0000 <mo>{</mo>\u0000 <mn>0</mn>\u0000 <mo>,</mo>\u0000 <mo>±</mo>\u0000 <mn>1</mn>\u0000 <mo>}</mo>\u0000 </mrow>\u0000 <annotation>$anotin lbrace 0,pm 1rbrace$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 4","pages":"1195-1220"},"PeriodicalIF":0.8,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12222","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50145833","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 the distribution of modular inverses from short intervals","authors":"Moubariz Z. Garaev,&nbsp;Igor E. Shparlinski","doi":"10.1112/mtk.12224","DOIUrl":"https://doi.org/10.1112/mtk.12224","url":null,"abstract":"<p>For a prime number <i>p</i> and integer <i>x</i> with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>gcd</mo>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>,</mo>\u0000 <mi>p</mi>\u0000 <mo>)</mo>\u0000 <mo>=</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$gcd (x,p)=1$</annotation>\u0000 </semantics></math>, let <math>\u0000 <semantics>\u0000 <mover>\u0000 <mi>x</mi>\u0000 <mo>¯</mo>\u0000 </mover>\u0000 <annotation>$overline{x}$</annotation>\u0000 </semantics></math> denote the multiplicative inverse of <i>x</i> modulo <i>p</i>. In this paper, we are interested in the problem of distribution modulo <i>p</i> of the sequence\u0000\u0000 </p><p>For any fixed <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mo>&gt;</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$A &gt; 1$</annotation>\u0000 </semantics></math> and for any sufficiently large integer <i>N</i>, there exists a prime number <i>p</i> with\u0000\u0000 </p><p>For any fixed positive <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>γ</mi>\u0000 <mo>&lt;</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$gamma &lt; 1$</annotation>\u0000 </semantics></math>, there exists a positive constant <i>c</i> such that the following holds: for any sufficiently large integer <i>N</i> there is a prime number <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>&gt;</mo>\u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation>$p &gt; N$</annotation>\u0000 </semantics></math> such that\u0000\u0000 </p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 4","pages":"1183-1194"},"PeriodicalIF":0.8,"publicationDate":"2023-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12224","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50145829","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":"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":"69 4","pages":"1145-1164"},"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":"69 4","pages":"1104-1144"},"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":"69 4","pages":"1081-1103"},"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":"69 4","pages":"1060-1080"},"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}