We classify maximal systems of arcs which intersect at most once on the 4-punctured sphere.
我们对在4点球面上最多相交一次的弧的极大系统进行了分类。
{"title":"Arcs intersecting at most once\u0000on the 4-punctured sphere","authors":"Pau‐Ling Tee","doi":"10.4064/cm8729-6-2023","DOIUrl":"https://doi.org/10.4064/cm8729-6-2023","url":null,"abstract":"We classify maximal systems of arcs which intersect at most once on the 4-punctured sphere.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42311147","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let G be a multiplicatively written finite group. We denote by E(G) the smallest integer t such that every sequence of t elements in G contains a product-one subsequence of length |G|. In 1961, Erdős, Ginzburg and Ziv proved that E(G) ≤ 2|G|−1 for every finite ablian group G and this result is known as the Erdős-Ginzburg-Ziv Theorem. In 2005, Zhuang and Gao conjectured that E(G) = d(G) + |G|, where d(G) is the small Davenport constant. In this paper, we confirm the conjecture for the case when G = 〈x, y|x = y = 1, xyx = y〉, where p is the smallest prime divisor of |G| and gcd(p(r − 1),m) = 1.
{"title":"On a conjecture of Zhuang and Gao","authors":"Yongke Qu, Yuanlin Li","doi":"10.4064/cm8685-2-2022","DOIUrl":"https://doi.org/10.4064/cm8685-2-2022","url":null,"abstract":"Let G be a multiplicatively written finite group. We denote by E(G) the smallest integer t such that every sequence of t elements in G contains a product-one subsequence of length |G|. In 1961, Erdős, Ginzburg and Ziv proved that E(G) ≤ 2|G|−1 for every finite ablian group G and this result is known as the Erdős-Ginzburg-Ziv Theorem. In 2005, Zhuang and Gao conjectured that E(G) = d(G) + |G|, where d(G) is the small Davenport constant. In this paper, we confirm the conjecture for the case when G = 〈x, y|x = y = 1, xyx = y〉, where p is the smallest prime divisor of |G| and gcd(p(r − 1),m) = 1.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48460074","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper gives new two-weight bump conditions for the sparse operators related to iterated commutators of fractional integrals. As applications, the two-weight bounds for iterated commutators of fractional integrals under more general bump conditions are obtained. Meanwhile, the necessity of two-weight bump conditions as well as the converse of Bloom type estimates for iterated commutators of fractional integrals are also given.
{"title":"Bump conditions and two-weight inequalities for commutators of fractional integrals","authors":"Yongming Wen, Huo-xiong Wu","doi":"10.4064/cm8703-4-2022","DOIUrl":"https://doi.org/10.4064/cm8703-4-2022","url":null,"abstract":"This paper gives new two-weight bump conditions for the sparse operators related to iterated commutators of fractional integrals. As applications, the two-weight bounds for iterated commutators of fractional integrals under more general bump conditions are obtained. Meanwhile, the necessity of two-weight bump conditions as well as the converse of Bloom type estimates for iterated commutators of fractional integrals are also given.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44459512","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
． 让s (n) = (cid): 80) d | n, d < n d合适divisors》denote the sum of n。自然是每到conjecture that for《equivalence n是整数k≥2,k th powerfree⇐⇒s (n)是k th powerfree珍藏几乎总是(asymptotic密度之意义,on a组1)。我们证明这个for k≥4。
{"title":"Powerfree sums of proper divisors","authors":"P. Pollack, A. Roy","doi":"10.4064/cm8616-10-2021","DOIUrl":"https://doi.org/10.4064/cm8616-10-2021","url":null,"abstract":". Let s ( n ) := (cid:80) d | n,d<n d denote the sum of the proper divisors of n . It is natural to conjecture that for each integer k ≥ 2 , the equivalence n is k th powerfree ⇐⇒ s ( n ) is k th powerfree holds almost always (meaning, on a set of asymptotic density 1 ). We prove this for k ≥ 4 .","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46099791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is known that G-functions solutions of a linear differential equation of order 1 with coefficients in Q(z) are algebraic (of a very precise form). No general result is known when the order is 2. In this paper, we determine the form of a G-function solution of an inhomogeneous equation of order 1 with coefficients in Q(z), as well as that of a G-function f of differential order 2 over Q(z) and such that f and f ′ are algebraically dependent over C(z). Our results apply more generally to holonomic Nilsson-Gevrey arithmetic series of order 0 that encompass G-functions.
{"title":"A note on $G$-operators of order $2$","authors":"S. Fischler, T. Rivoal","doi":"10.4064/cm8600-3-2022","DOIUrl":"https://doi.org/10.4064/cm8600-3-2022","url":null,"abstract":"It is known that G-functions solutions of a linear differential equation of order 1 with coefficients in Q(z) are algebraic (of a very precise form). No general result is known when the order is 2. In this paper, we determine the form of a G-function solution of an inhomogeneous equation of order 1 with coefficients in Q(z), as well as that of a G-function f of differential order 2 over Q(z) and such that f and f ′ are algebraically dependent over C(z). Our results apply more generally to holonomic Nilsson-Gevrey arithmetic series of order 0 that encompass G-functions.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46325998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this paper, we study the k -th moment of central values of the family of quadratic Dirichlet L -functions of moduli 8 p , with p ranging over odd primes. Assuming the truth of the generalized Riemann hypothesis, we establish sharp upper and lower bounds for the k -th power moment of these L -values for all real k ≥ 0.
． 本文研究了模为8p的二次Dirichlet L -函数族中心值的k阶矩，其中p的取值范围为奇数素数。在广义黎曼假设成立的前提下，我们为所有实k≥0时这些L值的k次幂矩建立了清晰的上界和下界。
{"title":"Bounds for moments of quadratic Dirichlet $L$-functions of prime-related moduli","authors":"Peng Gao, Liangyi Zhao","doi":"10.4064/cm8650-1-2022","DOIUrl":"https://doi.org/10.4064/cm8650-1-2022","url":null,"abstract":". In this paper, we study the k -th moment of central values of the family of quadratic Dirichlet L -functions of moduli 8 p , with p ranging over odd primes. Assuming the truth of the generalized Riemann hypothesis, we establish sharp upper and lower bounds for the k -th power moment of these L -values for all real k ≥ 0.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44228415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In spaces of metrics, we investigate topological distributions of the doubling property, the uniform disconnectedness, and the uniform perfectness, which are the quasi-symmetrically invariant properties appearing in the David--Semmes theorem. We show that the set of all doubling metrics and the set of all uniformly disconnected metrics are dense in spaces of metrics on finite-dimensional and zero-dimensional compact metrizable spaces, respectively. Conversely, this denseness of the sets implies the finite-dimensionality, zero-dimensionality, and the compactness of metrizable spaces. We also determine the topological distribution of the set of all uniformly perfect metrics in the space of metrics on the Cantor set.
{"title":"On dense subsets in spaces of metrics","authors":"Yoshito Ishiki","doi":"10.4064/cm8580-9-2021","DOIUrl":"https://doi.org/10.4064/cm8580-9-2021","url":null,"abstract":"In spaces of metrics, we investigate topological distributions of the doubling property, the uniform disconnectedness, and the uniform perfectness, which are the quasi-symmetrically invariant properties appearing in the David--Semmes theorem. We show that the set of all doubling metrics and the set of all uniformly disconnected metrics are dense in spaces of metrics on finite-dimensional and zero-dimensional compact metrizable spaces, respectively. Conversely, this denseness of the sets implies the finite-dimensionality, zero-dimensionality, and the compactness of metrizable spaces. We also determine the topological distribution of the set of all uniformly perfect metrics in the space of metrics on the Cantor set.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43096608","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We investigate the geometry of $C(K,X)$ and $ell_{infty}(X)$ spaces through complemented subspaces of the form $left(bigoplus_{iin varGamma}X_iright)_{c_0}$. Concerning the geometry of $C(K,X)$ spaces we extend some results of D. Alspach and E. M. Galego from cite{AlspachGalego}. On $ell_{infty}$-sums of Banach spaces we prove that if $ell_{infty}(X)$ has a complemented subspace isomorphic to $c_0(Y)$, then, for some $n in mathbb{N}$, $X^n$ has a subspace isomorphic to $c_0(Y)$. We further prove the following: (1) If $C(K)sim c_0(C(K))$ and $C(L)sim c_0(C(L))$ and $ell_{infty}(C(K))sim ell_{infty}(C(L))$, then $K$ and $L$ have the same cardinality. (2) If $K_1$ and $K_2$ are infinite metric compacta, then $ell_{infty}(C(K_1))sim ell_{infty}(C(K_2))$ if and only if $C(K_1)$ is isomorphic to $C(K_2)$.
我们通过$left(bigoplus_{iin varGamma}X_iright)_{c_0}$形式的互补子空间研究了$C(K,X)$和$ell_{infty}(X)$空间的几何。关于$C(K,X)$空间的几何性质，我们推广了D. Alspach和E. M. Galego在cite{AlspachGalego}上的一些结果。在Banach空间的$ell_{infty}$ -和上，证明了如果$ell_{infty}(X)$具有与$c_0(Y)$同构的补子空间，则对于某些$n in mathbb{N}$, $X^n$具有与$c_0(Y)$同构的子空间。我们进一步证明了:(1)如果$C(K)sim c_0(C(K))$与$C(L)sim c_0(C(L))$和$ell_{infty}(C(K))sim ell_{infty}(C(L))$，则$K$和$L$具有相同的基数。(2)如果$K_1$和$K_2$是无限度量紧致，则$ell_{infty}(C(K_1))sim ell_{infty}(C(K_2))$当且仅当$C(K_1)$同构于$C(K_2)$。
{"title":"Complementations in $C(K,X)$ and $ell _infty (X)$","authors":"Leandro Candido","doi":"10.4064/cm8868-10-2022","DOIUrl":"https://doi.org/10.4064/cm8868-10-2022","url":null,"abstract":"We investigate the geometry of $C(K,X)$ and $ell_{infty}(X)$ spaces through complemented subspaces of the form $left(bigoplus_{iin varGamma}X_iright)_{c_0}$. Concerning the geometry of $C(K,X)$ spaces we extend some results of D. Alspach and E. M. Galego from cite{AlspachGalego}. On $ell_{infty}$-sums of Banach spaces we prove that if $ell_{infty}(X)$ has a complemented subspace isomorphic to $c_0(Y)$, then, for some $n in mathbb{N}$, $X^n$ has a subspace isomorphic to $c_0(Y)$. We further prove the following: \u0000(1) If $C(K)sim c_0(C(K))$ and $C(L)sim c_0(C(L))$ and $ell_{infty}(C(K))sim ell_{infty}(C(L))$, then $K$ and $L$ have the same cardinality. \u0000(2) If $K_1$ and $K_2$ are infinite metric compacta, then $ell_{infty}(C(K_1))sim ell_{infty}(C(K_2))$ if and only if $C(K_1)$ is isomorphic to $C(K_2)$.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41882649","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We exhibit $88$ examples of rank zero elliptic curves over the rationals with $|sza(E)| > 63408^2$, which was the largest previously known value for any explicit curve. Our record is an elliptic curve $E$ with $|sza(E)| = 1029212^2 = 2^4cdot 79^2 cdot 3257^2$. We can use deep results by Kolyvagin, Kato, Skinner-Urban and Skinner to prove that, in some cases, these orders are the true orders of $sza$. For instance, $410536^2$ is the true order of $sza(E)$ for $E= E_4(21,-233)$ from the table in section 2.3.
{"title":"Elliptic curves with exceptionally large analytic order of the Tate–Shafarevich groups","authors":"A. Dkabrowski, L. Szymaszkiewicz","doi":"10.4064/CM8008-9-2020","DOIUrl":"https://doi.org/10.4064/CM8008-9-2020","url":null,"abstract":"We exhibit $88$ examples of rank zero elliptic curves over the rationals with $|sza(E)| > 63408^2$, which was the largest previously known value for any explicit curve. Our record is an elliptic curve $E$ with $|sza(E)| = 1029212^2 = 2^4cdot 79^2 cdot 3257^2$. We can use deep results by Kolyvagin, Kato, Skinner-Urban and Skinner to prove that, in some cases, these orders are the true orders of $sza$. For instance, $410536^2$ is the true order of $sza(E)$ for $E= E_4(21,-233)$ from the table in section 2.3.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46148455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that any one-relator group $G$ is the fundamental group of a compact Sasakian manifold if and only if $G$ is either finite cyclic or isomorphic to the fundamental group of a compact Riemann surface of genus g > 0 with at most one orbifold point of order $n geq 1$. We also classify all groups of deficiency at least two that are also the fundamental group of some compact Sasakian manifold.
{"title":"One-relator Sasakian groups","authors":"I. Biswas, Mahan Mj","doi":"10.4064/CM8521-3-2021","DOIUrl":"https://doi.org/10.4064/CM8521-3-2021","url":null,"abstract":"We prove that any one-relator group $G$ is the fundamental group of a compact Sasakian manifold if and only if $G$ is either finite cyclic or isomorphic to the fundamental group of a compact Riemann surface of genus g > 0 with at most one orbifold point of order $n geq 1$. We also classify all groups of deficiency at least two that are also the fundamental group of some compact Sasakian manifold.","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48612998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：