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Moscow Journal of Combinatorics and Number Theory最新文献

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Square-full primitive roots in arithmetic progressions 算术级数中的平方全原始根
Moscow Journal of Combinatorics and Number Theory Pub Date : 2020-08-07 DOI: 10.2140/moscow.2020.9.187
V. Laohakosol, T. Srichan, Pinthira Tangsupphathawat
{"title":"Square-full primitive roots in arithmetic progressions","authors":"V. Laohakosol, T. Srichan, Pinthira Tangsupphathawat","doi":"10.2140/moscow.2020.9.187","DOIUrl":"https://doi.org/10.2140/moscow.2020.9.187","url":null,"abstract":"An asymptotic estimate for the number of positive primitive roots which are square-full integers in arithmetic progressions is derived. The employed method combines two techniques and is based on the character-sum method involving two characters; one character is to take care of being a primitive root, based on a result of Shapiro, and the other character is to take care of being square-full, based on a result of Munsch.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/moscow.2020.9.187","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49511627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Alon–Tarsi numbers of direct products 直接产品的Alon–Tarsi数
Moscow Journal of Combinatorics and Number Theory Pub Date : 2020-07-14 DOI: 10.2140/moscow.2021.10.271
A. Gordeev, F. Petrov
{"title":"Alon–Tarsi numbers of direct products","authors":"A. Gordeev, F. Petrov","doi":"10.2140/moscow.2021.10.271","DOIUrl":"https://doi.org/10.2140/moscow.2021.10.271","url":null,"abstract":"We provide a general framework on the coefficients of the graph polynomials of graphs which are Cartesian products. As a corollary, we prove that if $G=(V,E)$ is a graph with degrees of vertices $2d(v), vin V$, and the graph polynomial $prod_{(i,j)in E} (x_j-x_i)$ contains an ``almost central'' monomial (that means a monomial $prod_v x_v^{c_v}$, where $|c_v-d(v)|leqslant 1$ for all $vin V$), then the Cartesian product $Gsquare C_{2n}$ is $(d(cdot)+2)$-choosable.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48327890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the coefficient-choosing game 关于系数选择博弈
Moscow Journal of Combinatorics and Number Theory Pub Date : 2020-07-01 DOI: 10.2140/moscow.2021.10.183
Divyum Sharma, L. Singhal
{"title":"On the coefficient-choosing game","authors":"Divyum Sharma, L. Singhal","doi":"10.2140/moscow.2021.10.183","DOIUrl":"https://doi.org/10.2140/moscow.2021.10.183","url":null,"abstract":"Nora and Wanda are two players who choose coefficients of a degree $d$ polynomial from some fixed unital commutative ring $R$. Wanda is declared the winner if the polynomial has a root in the ring of fractions of $R$ and Nora is declared the winner otherwise. We extend the theory of these games given by Gasarch, Washington and Zbarsky to all finite cyclic rings and determine the possible outcomes. A family of examples is also constructed using discrete valuation rings for a variant of the game proposed by these authors.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46908895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Shattered matchings in intersecting hypergraphs 相交超图中的破碎匹配
Moscow Journal of Combinatorics and Number Theory Pub Date : 2020-05-11 DOI: 10.2140/MOSCOW.2021.10.49
P. Frankl, J. Pach
{"title":"Shattered matchings in intersecting hypergraphs","authors":"P. Frankl, J. Pach","doi":"10.2140/MOSCOW.2021.10.49","DOIUrl":"https://doi.org/10.2140/MOSCOW.2021.10.49","url":null,"abstract":"Let $X$ be an $n$-element set, where $n$ is even. We refute a conjecture of J. Gordon and Y. Teplitskaya, according to which, for every maximal intersecting family $mathcal{F}$ of $frac{n}2$-element subsets of $X$, one can partition $X$ into $frac{n}2$ disjoint pairs in such a way that no matter how we pick one element from each of the first $frac{n}2 - 1$ pairs, the set formed by them can always be completed to a member of $mathcal{F}$ by adding an element of the last pair. \u0000The above problem is related to classical questions in extremal set theory. For any $tge 2$, we call a family of sets $mathcal{F}subset 2^X$ {em $t$-separable} if for any ordered pair of elements $(x,y)$ of $X$, there exists $Finmathcal{F}$ such that $Fcap{x,y}={x}$. For a fixed $t, 2le tle 5$ and $nrightarrowinfty$, we establish asymptotically tight estimates for the smallest integer $s=s(n,t)$ such that every family $mathcal{F}$ with $|mathcal{F}|ge s$ is $t$-separable.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45964000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A gap of the exponents of repetitions of Sturmian words 斯图尔曼语单词重复指数的差距
Moscow Journal of Combinatorics and Number Theory Pub Date : 2020-03-24 DOI: 10.2140/moscow.2021.10.203
Suzue Ohnaka, Takao Watanabe
{"title":"A gap of the exponents of repetitions of Sturmian words","authors":"Suzue Ohnaka, Takao Watanabe","doi":"10.2140/moscow.2021.10.203","DOIUrl":"https://doi.org/10.2140/moscow.2021.10.203","url":null,"abstract":"By measuring the smallest second occuring time of every factor of an infinite word $x$, Bugeaud and Kim introduced a new quantity ${rm rep}(x)$ called the exponent of repetition of $x$. Among other results, Bugeaud and Kim proved that $1 leq {rm rep}(x) leq r_{max} = sqrt{10} - 3/2$ and $r_{max}$ is the isolated maximum value when $x$ varies over the Sturmian words. In this paper, we determine the value $r_1$ such that there is no Sturmian word $x$ satisfying $r_1 < {rm rep}(x) < r_{max}$ and $r_1$ is an accumulate point of the set of ${rm rep}(x)$ when $x$ runs over the Sturmian words.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43915226","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Bounding the difference of two singular moduli 限定两个奇异模的差
Moscow Journal of Combinatorics and Number Theory Pub Date : 2020-03-12 DOI: 10.2140/moscow.2021.10.95
Yulin Cai
{"title":"Bounding the difference of two singular moduli","authors":"Yulin Cai","doi":"10.2140/moscow.2021.10.95","DOIUrl":"https://doi.org/10.2140/moscow.2021.10.95","url":null,"abstract":"For a fixed singular modulus $alpha$, we give an effective upper bound of norm of $x-alpha$ for another singular modulus $x$ with large discriminant.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44241443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Bounded remainder sets for rotations on higher-dimensional adelic tori 高维双曲面上旋转的有界余集
Moscow Journal of Combinatorics and Number Theory Pub Date : 2020-02-10 DOI: 10.2140/moscow.2021.10.111
A. Das, Joanna Furno, A. Haynes
{"title":"Bounded remainder sets for rotations on higher-dimensional adelic tori","authors":"A. Das, Joanna Furno, A. Haynes","doi":"10.2140/moscow.2021.10.111","DOIUrl":"https://doi.org/10.2140/moscow.2021.10.111","url":null,"abstract":"In this paper we give a simple, explicit construction of polytopal bounded remainder sets of all possible volumes, for any irrational rotation on the $d$ dimensional adelic torus $mathbb{A}^d/mathbb{Q}^d$. Our construction involves ideas from dynamical systems and harmonic analysis on the adeles, as well as a geometric argument that reduces the existence argument to the case of an irrational rotation on the torus $mathbb{R}^d/mathbb{Q}^d$.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45746532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the roots of the Poupard and Kreweras polynomials 关于Poupard和Kreweras多项式的根
Moscow Journal of Combinatorics and Number Theory Pub Date : 2020-01-06 DOI: 10.2140/moscow.2020.9.163
F. Chapoton, Guo-Niu Han
{"title":"On the roots of the Poupard and Kreweras polynomials","authors":"F. Chapoton, Guo-Niu Han","doi":"10.2140/moscow.2020.9.163","DOIUrl":"https://doi.org/10.2140/moscow.2020.9.163","url":null,"abstract":"The Poupard polynomials are polynomials in one variable with integer coefficients, with some close relationship to Bernoulli and tangent numbers. They also have a combinatorial interpretation. We prove that every Poupard polynomial has all its roots on the unit circle. We also obtain the same property for another sequence of polynomials introduced by Kreweras and related to Genocchi numbers. This is obtained through a general statement about some linear operators acting on palindromic polynomials.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43310892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Can polylogarithms at algebraic points be linearly independent? 代数点上的多对数是线性无关的吗?
Moscow Journal of Combinatorics and Number Theory Pub Date : 2019-12-09 DOI: 10.2140/moscow.2020.9.389
Sinnou David, Noriko Hirata-Kohno, M. Kawashima
{"title":"Can polylogarithms at algebraic points be linearly independent?","authors":"Sinnou David, Noriko Hirata-Kohno, M. Kawashima","doi":"10.2140/moscow.2020.9.389","DOIUrl":"https://doi.org/10.2140/moscow.2020.9.389","url":null,"abstract":"Let $r,m$ be positive integers. Let $0le x <1$ be a rational number. Let $Phi_s(x,z)$ be the $s$-th Lerch function $sum_{k=0}^{infty}tfrac{z^{k+1}}{(k+x+1)^s}$ with $s=1,2,ldots ,r$. When $x=0$, this is the polylogarithmic function. Let $alpha_1,ldots ,alpha_m$ be pairwise distinct algebraic numbers with $0<|alpha_j|<1$ $(1 le j le m)$. In this article, we state a linear independence criterion over algebraic number fields of all the $rm+1$ numbers $:$ $Phi_1(x,alpha_1),Phi_2(x,alpha_1),ldots, Phi_r(x,alpha_1),Phi_1(x,alpha_2),Phi_2(x,alpha_2),ldots, Phi_r(x,alpha_2),ldots,Phi_1(x,alpha_m),Phi_2(x,alpha_m),ldots, Phi_r(x,alpha_m)$ and $1$. This is the first result that gives a sufficient condition for the linear independence of values of the $r$ Lerch functions $Phi_1(x,z),Phi_2(x,z),ldots, Phi_r(x,z)$ at $m$ distinct algebraic points without any assumption for $r$ and $m$, even for the case $x=0$, the polylogarithms. We give an outline of our proof and explain basic idea.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46181340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 9
On transcendental entire functions with infinitely many derivatives taking integer values at several points 关于具有无穷多个导数在若干点取整数值的超越整函数
Moscow Journal of Combinatorics and Number Theory Pub Date : 2019-11-30 DOI: 10.2140/MOSCOW.2020.9.371
M. Waldschmidt
{"title":"On transcendental entire functions with infinitely many derivatives taking integer values at several points","authors":"M. Waldschmidt","doi":"10.2140/MOSCOW.2020.9.371","DOIUrl":"https://doi.org/10.2140/MOSCOW.2020.9.371","url":null,"abstract":"Let $s_0,s_1,dots,s_{m-1}$ be complex numbers and $r_0,dots,r_{m-1}$ rational integers in the range $0le r_jle m-1$. Our first goal is to prove that if an entire function $f$ of sufficiently small exponential type satisfies $f^{(mn+r_j)}(s_j)in{mathbb Z}$ for $0le jle m-1$ and all sufficiently large $n$, then $f$ is a polynomial. Under suitable assumptions on $s_0,s_1,dots,s_{m-1}$ and $r_0,dots,r_{m-1}$, we introduce interpolation polynomials $Lambda_{nj}$, ($nge 0$, $0le jle m-1$) satisfying $$ Lambda_{nj}^{(mk+r_ell)}(s_ell)=delta_{jell}delta_{nk}, quadhbox{for}quad n, kge 0 quadhbox{and}quad 0le j, ellle m-1 $$ and we show that any entire function $f$ of sufficiently small exponential type has a convergent expansion $$ f(z)=sum_{nge 0} sum_{j=0}^{m-1}f^{(mn+r_j)}(s_j)Lambda_{nj}(z). $$ The case $r_j=j$ for $0le jle m-1$ involves successive derivatives $f^{(n)}(w_n)$ of $f$ evaluated at points of a periodic sequence ${mathbf{w}}=(w_n)_{nge 0}$ of complex numbers, where $w_{mh+j}=s_j$ ($hge 0$, $0le jle m$). More generally, given a bounded (not necessarily periodic) sequence ${mathbf{w}}=(w_n)_{nge 0}$ of complex numbers, we consider similar interpolation formulae $$ f(z)=sum_{nge 0}f^{(n)}(w_n)Omega_{mathbf{w},n}(z) $$ involving polynomials $Omega_{mathbf{w},n}(z)$ which were introduced by W.~Gontcharoff in 1930. Under suitable assumptions, we show that the hypothesis $f^{(n)}(w_n)in{mathbb Z}$ for all sufficiently large $n$ implies that $f$ is a polynomial.","PeriodicalId":36590,"journal":{"name":"Moscow Journal of Combinatorics and Number Theory","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46473929","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
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