Mathematika最新文献

{"title":"On a geometric combination of functions related to Prékopa–Leindler inequality","authors":"Graziano Crasta,&nbsp;Ilaria Fragalà","doi":"10.1112/mtk.12192","DOIUrl":"10.1112/mtk.12192","url":null,"abstract":"<p>We introduce a new operation between nonnegative integrable functions on <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>$mathbb {R}^n$</annotation>\u0000 </semantics></math>, that we call <i>geometric combination</i>; it is obtained via a mass transportation approach, playing with inverse distribution functions. The main feature of this operation is that the Lebesgue integral of the geometric combination equals the geometric mean of the two separate integrals; as a natural consequence, we derive a new functional inequality of Prékopa–Leindler type. When applied to the characteristic functions of two measurable sets, their geometric combination provides a set whose volume equals the geometric mean of the two separate volumes. In the framework of convex bodies, by comparing the geometric combination with the 0-sum, we get an alternative proof of the log-Brunn–Minkowski inequality for unconditional convex bodies and for convex bodies with <i>n</i> symmetries.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 2","pages":"482-507"},"PeriodicalIF":0.8,"publicationDate":"2023-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12192","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46898849","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":"Unions of lines in \u0000 \u0000 \u0000 R\u0000 n\u0000 \u0000 $mathbb {R}^n$","authors":"Joshua Zahl","doi":"10.1112/mtk.12190","DOIUrl":"10.1112/mtk.12190","url":null,"abstract":"<p>We prove a conjecture of D. Oberlin on the dimension of unions of lines in <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>$mathbb {R}^n$</annotation>\u0000 </semantics></math>. If <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>⩾</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$dgeqslant 1$</annotation>\u0000 </semantics></math> is an integer, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>0</mn>\u0000 <mo>⩽</mo>\u0000 <mi>β</mi>\u0000 <mo>⩽</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$0leqslant beta leqslant 1$</annotation>\u0000 </semantics></math>, and <i>L</i> is a set of lines in <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>$mathbb {R}^n$</annotation>\u0000 </semantics></math> with Hausdorff dimension at least <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mo>(</mo>\u0000 <mi>d</mi>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 <mo>+</mo>\u0000 <mi>β</mi>\u0000 </mrow>\u0000 <annotation>$2(d-1)+beta$</annotation>\u0000 </semantics></math>, then the union of the lines in <i>L</i> has Hausdorff dimension at least <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>+</mo>\u0000 <mi>β</mi>\u0000 </mrow>\u0000 <annotation>$d + beta$</annotation>\u0000 </semantics></math>. Our proof combines a refined version of the multilinear Kakeya theorem by Carbery and Valdimarsson with the multilinear → linear argument of Bourgain and Guth.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 2","pages":"473-481"},"PeriodicalIF":0.8,"publicationDate":"2023-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12190","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44163916","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":"Affine subspace concentration conditions for centered polytopes","authors":"Ansgar Freyer,&nbsp;Martin Henk,&nbsp;Christian Kipp","doi":"10.1112/mtk.12189","DOIUrl":"10.1112/mtk.12189","url":null,"abstract":"<p>Recently, K.-Y. Wu introduced affine subspace concentration conditions for the cone volumes of polytopes and proved that the cone volumes of centered, reflexive, smooth lattice polytopes satisfy these conditions. We extend the result to arbitrary centered polytopes.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 2","pages":"458-472"},"PeriodicalIF":0.8,"publicationDate":"2023-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12189","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45466250","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 Borsuk–Ulam theorems and convex sets","authors":"M. C. Crabb","doi":"10.1112/mtk.12186","DOIUrl":"10.1112/mtk.12186","url":null,"abstract":"<p>The Intermediate Value Theorem is used to give an elementary proof of a Borsuk–Ulam theorem of Adams, Bush and Frick [1] that if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mo>:</mo>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 <mo>→</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mi>k</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$f: S^1rightarrow {mathbb {R}}^{2k+1}$</annotation>\u0000 </semantics></math> is a continuous function on the unit circle <i>S</i>¹ in <math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>${mathbb {C}}$</annotation>\u0000 </semantics></math> such that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mo>(</mo>\u0000 <mo>−</mo>\u0000 <mi>z</mi>\u0000 <mo>)</mo>\u0000 <mo>=</mo>\u0000 <mo>−</mo>\u0000 <mi>f</mi>\u0000 <mo>(</mo>\u0000 <mi>z</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$f(-z)=-f(z)$</annotation>\u0000 </semantics></math> for all <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>z</mi>\u0000 <mo>∈</mo>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$zin S^1$</annotation>\u0000 </semantics></math>, then there is a finite subset <i>X</i> of <i>S</i>¹ of diameter at most <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>π</mi>\u0000 <mo>−</mo>\u0000 <mi>π</mi>\u0000 <mo>/</mo>\u0000 <mo>(</mo>\u0000 <mn>2</mn>\u0000 <mi>k</mi>\u0000 <mo>+</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$pi -pi /(2k+1)$</annotation>\u0000 </semantics></math> (in the standard metric in which the circle has circumference of length 2π) such the convex hull of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mo>(</mo>\u0000 <mi>X</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$f(X)$</annotation>\u0000 </semantics></math> contains <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>0</mn>\u0000 <mo>∈</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 ","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 2","pages":"366-370"},"PeriodicalIF":0.8,"publicationDate":"2023-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41522426","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":"Extremal values of semi-regular continuants and codings of interval exchange transformations","authors":"Alessandro De Luca,&nbsp;Marcia Edson,&nbsp;Luca Q. Zamboni","doi":"10.1112/mtk.12185","DOIUrl":"10.1112/mtk.12185","url":null,"abstract":"<p>Given a set <math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$mathbb {A}$</annotation>\u0000 </semantics></math> consisting of positive integers <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>a</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>&lt;</mo>\u0000 <msub>\u0000 <mi>a</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <mo>&lt;</mo>\u0000 <mi>⋯</mi>\u0000 <mo>&lt;</mo>\u0000 <msub>\u0000 <mi>a</mi>\u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$a_1&lt;a_2&lt;cdots &lt;a_k$</annotation>\u0000 </semantics></math> and a <i>k</i>-term partition <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>P</mi>\u0000 <mo>:</mo>\u0000 <msub>\u0000 <mi>n</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>+</mo>\u0000 <msub>\u0000 <mi>n</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <mo>+</mo>\u0000 <mi>⋯</mi>\u0000 <mo>+</mo>\u0000 <msub>\u0000 <mi>n</mi>\u0000 <mi>k</mi>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation>$P: n_1+n_2 + cdots + n_k=n$</annotation>\u0000 </semantics></math>, find the extremal denominators of the regular and semi-regular continued fraction <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mn>0</mn>\u0000 <mo>;</mo>\u0000 <msub>\u0000 <mi>x</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>x</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mtext>…</mtext>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>x</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <annotation>$[0;x_1,x_2,ldots ,x_n]$</annotation>\u0000 </semantics></math> with partial quotients <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>x</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 <mo>∈</mo>\u0000 <mi>A</mi>\u0000 </mrow>\u0000 <annotation>$x_iin mathbb {A}$</annotation>\u0000 </semantics></math> and where each <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>a</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 <annotation>$","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 2","pages":"432-457"},"PeriodicalIF":0.8,"publicationDate":"2023-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48323863","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":"The distribution of geodesics on the cube surface","authors":"Yuxuan Yang","doi":"10.1112/mtk.12188","DOIUrl":"10.1112/mtk.12188","url":null,"abstract":"<p>We establish a Kronecker–Weyl type result, on time-quantitative equidistribution for a natural non-integrable system, geodesic flow on the cube surface. Our tool is the shortline-ancestor method developed in Beck, Donders, and Yang [Acta Math. Hungar. <b>161</b> (2020), 66–184] and Beck, Donders, and Yang [Acta Math. Hungar. <i>162</i> (2020), 220–324], modified in an appropriate way to embrace all slopes. The method is further enhanced by the symmetry of the cube through the use of the irreducible representations of the symmetric group <i>S</i>₄ which makes the determination of the irregularity exponent substantially simpler.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 2","pages":"371-431"},"PeriodicalIF":0.8,"publicationDate":"2023-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49016445","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":"Average shadowing and gluing property","authors":"Michael Blank","doi":"10.1112/mtk.12187","DOIUrl":"10.1112/mtk.12187","url":null,"abstract":"<p>The purpose of this work is threefold: (i) extend shadowing theory for discontinuous and non-invertible systems, (ii) consider more general classes of perturbations (for example, small only on average), (iii) establish a general theory based on the property that the shadowing holds for the case of a single perturbation. The “gluing” construction used in the analysis of the last property turns out to be the key point of this theory.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 2","pages":"349-365"},"PeriodicalIF":0.8,"publicationDate":"2023-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46323401","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 number variance of zeta zeros and a conjecture of Berry","authors":"Meghann Moriah Lugar,&nbsp;Micah B. Milinovich,&nbsp;Emily Quesada-Herrera","doi":"10.1112/mtk.12184","DOIUrl":"10.1112/mtk.12184","url":null,"abstract":"<p>Assuming the Riemann hypothesis, we prove estimates for the variance of the real and imaginary part of the logarithm of the Riemann zeta function in short intervals. We give three different formulations of these results. Assuming a conjecture of Chan for how often gaps between zeros can be close to a fixed non-zero value, we prove a conjecture of Berry (1988) for the number variance of zeta zeros in the non-universal regime. In this range, Gaussian unitary ensemble statistics do not describe the distribution of the zeros. We also calculate lower order terms in the second moment of the logarithm of the modulus of the Riemann zeta function on the critical line. Assuming Montgomery's pair correlation conjecture, this establishes a special case of a conjecture of Keating and Snaith (2000).</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 2","pages":"303-348"},"PeriodicalIF":0.8,"publicationDate":"2023-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12184","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47571627","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 strong chains of sets and functions","authors":"Tanmay C. Inamdar","doi":"10.1112/mtk.12183","DOIUrl":"10.1112/mtk.12183","url":null,"abstract":"<p>Shelah has shown that there are no chains of length ω₃ increasing modulo finite in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mrow></mrow>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </msup>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>${}^{omega _2}omega _2$</annotation>\u0000 </semantics></math>. We improve this result to sets. That is, we show that there are no chains of length ω₃ in <math>\u0000 <semantics>\u0000 <msup>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 </msup>\u0000 <annotation>$[omega _2]^{aleph _2}$</annotation>\u0000 </semantics></math> increasing modulo finite. This contrasts with results of Koszmider who has shown that there are, consistently, chains of length ω₂ increasing modulo finite in <math>\u0000 <semantics>\u0000 <msup>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 </msup>\u0000 <annotation>$[omega _1]^{aleph _1}$</annotation>\u0000 </semantics></math> as well as in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mrow></mrow>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 </msup>\u0000 <msub>\u0000 <mi>ω</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>${}^{omega _1}omega _1$</annotation>\u0000 </semantics></math>. More generally, we study the depth of function spaces <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mrow></mrow>\u0000 <mi>κ</mi>\u0000 </msup>\u0000 <mi>μ</mi>\u0000 </mrow>\u0000 <annotation>${}^kappa mu$</annotation>\u0000 ","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 1","pages":"286-301"},"PeriodicalIF":0.8,"publicationDate":"2023-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12183","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48463431","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":"Discorrelation of multiplicative functions with nilsequences and its application on coefficients of automorphic L-functions","authors":"Xiaoguang He,&nbsp;Mengdi Wang","doi":"10.1112/mtk.12182","DOIUrl":"10.1112/mtk.12182","url":null,"abstract":"<p>We introduce a class of multiplicative functions in which each function satisfies some statistic conditions, and then prove that the above functions are not correlated with finite degree polynomial nilsequences. Besides, we give two applications of this result. One is that the twisting of coefficients of automorphic <i>L</i>-function on <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <msub>\u0000 <mi>L</mi>\u0000 <mi>m</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>m</mi>\u0000 <mo>⩾</mo>\u0000 <mn>2</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$GL_m (m geqslant 2)$</annotation>\u0000 </semantics></math> and polynomial nilsequences has logarithmic decay; the other is that the mean value of the Möbius function, coefficients of automorphic <i>L</i>-function, and polynomial nilsequences also has logarithmic decay.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":"69 1","pages":"250-285"},"PeriodicalIF":0.8,"publicationDate":"2022-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12182","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48068428","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}