Mathematika最新文献

{"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":null,"pages":null},"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":null,"pages":null},"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}

{"title":"Exponentially larger affine and projective caps","authors":"Christian Elsholtz,&nbsp;Gabriel F. Lipnik","doi":"10.1112/mtk.12173","DOIUrl":"10.1112/mtk.12173","url":null,"abstract":"<p>In spite of a recent breakthrough on upper bounds of the size of cap sets (by Croot, Lev and Pach and by Ellenberg and Gijswijt), the classical cap set constructions had not been affected. In this work, we introduce a very different method of construction for caps in all affine spaces with odd prime modulus <i>p</i>. Moreover, we show that for all primes <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>≡</mo>\u0000 <mn>5</mn>\u0000 <mspace></mspace>\u0000 <mi>mod</mi>\u0000 <mspace></mspace>\u0000 <mn>6</mn>\u0000 </mrow>\u0000 <annotation>$p equiv 5 bmod 6$</annotation>\u0000 </semantics></math> with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>⩽</mo>\u0000 <mn>41</mn>\u0000 </mrow>\u0000 <annotation>$p leqslant 41$</annotation>\u0000 </semantics></math>, the new construction leads to an exponentially larger growth of the affine and projective caps in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>AG</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mi>p</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${rm AG}(n,p)$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>PG</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>,</mo>\u0000 <mi>p</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>${rm PG}(n,p)$</annotation>\u0000 </semantics></math>. For example, when <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 <mo>=</mo>\u0000 <mn>23</mn>\u0000 </mrow>\u0000 <annotation>$p=23$</annotation>\u0000 </semantics></math>, the existence of caps with growth <math>\u0000 <semantics>\u0000 <msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mn>8.0875</mn>\u0000 <mtext>…</mtext>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>$(8.0875ldots )^n$</annotation>\u0000 </semantics></math> follows from a three-dimensional example of Bose, and the only improvement had been to <math>\u0000 <semantics>\u0000 <msup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mn>8.0901</mn>\u0000 <mtext>…</mtext>\u0000 <mo>)</mo>\u0000 </m","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12173","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"9390986","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":"Correlations of multiplicative functions in function fields","authors":"Oleksiy Klurman,&nbsp;Alexander P. Mangerel,&nbsp;Joni Teräväinen","doi":"10.1112/mtk.12181","DOIUrl":"10.1112/mtk.12181","url":null,"abstract":"<p>We develop an approach to study character sums, weighted by a multiplicative function <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mo>:</mo>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mi>q</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mi>t</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <mo>→</mo>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>1</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$fcolon mathbb {F}_q[t]rightarrow S^1$</annotation>\u0000 </semantics></math>, of the form\u0000\u0000 </p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12181","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48206343","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":"Geometric generalizations of the square sieve, with an application to cyclic covers","authors":"Alina Bucur,&nbsp;Alina Carmen Cojocaru,&nbsp;Matilde N. Lalín,&nbsp;Lillian B. Pierce","doi":"10.1112/mtk.12180","DOIUrl":"10.1112/mtk.12180","url":null,"abstract":"<p>We formulate a general problem: Given projective schemes <math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$mathbb {Y}$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$mathbb {X}$</annotation>\u0000 </semantics></math> over a global field <i>K</i> and a <i>K</i>-morphism η from <math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$mathbb {Y}$</annotation>\u0000 </semantics></math> to <math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$mathbb {X}$</annotation>\u0000 </semantics></math> of finite degree, how many points in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 <mo>(</mo>\u0000 <mi>K</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathbb {X}(K)$</annotation>\u0000 </semantics></math> of height at most <i>B</i> have a pre-image under η in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Y</mi>\u0000 <mo>(</mo>\u0000 <mi>K</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$mathbb {Y}(K)$</annotation>\u0000 </semantics></math>? This problem is inspired by a well-known conjecture of Serre on quantitative upper bounds for the number of points of bounded height on an irreducible projective variety defined over a number field. We give a nontrivial answer to the general problem when <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>K</mi>\u0000 <mo>=</mo>\u0000 <msub>\u0000 <mi>F</mi>\u0000 <mi>q</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>T</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$K=mathbb {F}_q(T)$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <mi>Y</mi>\u0000 <annotation>$mathbb {Y}$</annotation>\u0000 </semantics></math> is a prime degree cyclic cover of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>X</mi>\u0000 <mo>=</mo>\u0000 <msubsup>\u0000 <mi>P</mi>\u0000 <mi>K</mi>\u0000 <mi>n</mi>\u0000 </msubsup>\u0000 </mrow>\u0000 <annotation>$mathbb {X}=mathbb {P}_{K}^n$</annotation>\u0000 </semantics></math>. Our tool is a new geometric sieve, which generalizes the polynomial sieve to a geometric setting over global function fields.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48518388","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":"Discrete isoperimetric problems in spaces of constant curvature","authors":"Bushra Basit,&nbsp;Zsolt Lángi","doi":"10.1112/mtk.12175","DOIUrl":"10.1112/mtk.12175","url":null,"abstract":"<p>The aim of this paper is to prove isoperimetric inequalities for simplices and polytopes with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>+</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$d+2$</annotation>\u0000 </semantics></math> vertices in Euclidean, spherical and hyperbolic <i>d</i>-space. In particular, we find the minimal volume <i>d</i>-dimensional hyperbolic simplices and spherical tetrahedra of a given inradius. Furthermore, we investigate the properties of maximal volume spherical and hyperbolic polytopes with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>+</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$d+2$</annotation>\u0000 </semantics></math> vertices with a given circumradius, and the hyperbolic polytopes with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>+</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$d+2$</annotation>\u0000 </semantics></math> vertices with a given inradius and having a minimal volume or minimal total edge length. Finally, for any <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>⩽</mo>\u0000 <mi>k</mi>\u0000 <mo>⩽</mo>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation>$1 leqslant k leqslant d$</annotation>\u0000 </semantics></math>, we investigate the properties of Euclidean simplices and polytopes with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>+</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$d+2$</annotation>\u0000 </semantics></math> vertices having a fixed inradius and a minimal volume of its <i>k</i>-skeleton. The main tool of our investigation is Euclidean, spherical and hyperbolic Steiner symmetrization.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12175","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43250441","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 very short proof of the functional equation for ζ","authors":"Keith Ball","doi":"10.1112/mtk.12172","DOIUrl":"10.1112/mtk.12172","url":null,"abstract":"<p>This article contains a 3/4-page proof of the functional equation for Riemann's ζ function.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/mtk.12172","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48684566","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":"Quadratic characters with positive partial sums","authors":"A. B. Kalmynin","doi":"10.1112/mtk.12179","DOIUrl":"10.1112/mtk.12179","url":null,"abstract":"<p>Let <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mo>+</mo>\u0000 </msup>\u0000 <annotation>$mathcal{L}^+$</annotation>\u0000 </semantics></math> be the set of all primes <i>p</i> for which the sums of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mfrac>\u0000 <mi>n</mi>\u0000 <mi>p</mi>\u0000 </mfrac>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(frac{n}{p})$</annotation>\u0000 </semantics></math> over the interval [1, <i>N</i>] are non-negative for all <i>N</i>. We prove that the estimate\u0000\u0000 </p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45548160","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":"Irregularities of distribution for bounded sets and half-spaces","authors":"Luca Brandolini,&nbsp;Leonardo Colzani,&nbsp;Giancarlo Travaglini","doi":"10.1112/mtk.12178","DOIUrl":"10.1112/mtk.12178","url":null,"abstract":"<p>We prove a general result on irregularities of distribution for Borel sets intersected with bounded measurable sets or affine half-spaces.</p>","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41430967","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":"Permutations and the divisor graph of [1, n]","authors":"Nathan McNew","doi":"10.1112/mtk.12177","DOIUrl":"10.1112/mtk.12177","url":null,"abstract":"<p>Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>S</mi>\u0000 <mi>div</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$S_{rm div}(n)$</annotation>\u0000 </semantics></math> denote the set of permutations π of <i>n</i> such that for each <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>⩽</mo>\u0000 <mi>j</mi>\u0000 <mo>⩽</mo>\u0000 <mi>n</mi>\u0000 </mrow>\u0000 <annotation>$1leqslant j leqslant n$</annotation>\u0000 </semantics></math> either <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>j</mi>\u0000 <mo>∣</mo>\u0000 <mi>π</mi>\u0000 <mo>(</mo>\u0000 <mi>j</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$j mid pi (j)$</annotation>\u0000 </semantics></math> or <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>π</mi>\u0000 <mo>(</mo>\u0000 <mi>j</mi>\u0000 <mo>)</mo>\u0000 <mo>∣</mo>\u0000 <mi>j</mi>\u0000 </mrow>\u0000 <annotation>$pi (j) mid j$</annotation>\u0000 </semantics></math>. These permutations can also be viewed as vertex-disjoint directed cycle covers of the divisor graph <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>D</mi>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mi>n</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 </msub>\u0000 <annotation>$mathcal {D}_{[1,n]}$</annotation>\u0000 </semantics></math> on vertices <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>v</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mtext>…</mtext>\u0000 <mo>,</mo>\u0000 <msub>\u0000 <mi>v</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$v_1, ldots , v_n$</annotation>\u0000 </semantics></math> with an edge between <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>v</mi>\u0000 <mi>i</mi>\u0000 </msub>\u0000 <annotation>$v_i$</annotation>\u0000 </semantics></math> and <math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>v</mi>\u0000 <mi>j</mi>\u0000 </msub>\u0000 <annotation>$v_j$</annotation>\u0000 </semantics></math> if <math>\u0000 <semantics>\u0000 ","PeriodicalId":18463,"journal":{"name":"Mathematika","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45497519","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}