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On the sum of a prime power and a power in short intervals 求质数幂和短间隔内幂的和
4区 数学
Arkiv for Matematik Pub Date : 2023-01-01 DOI: 10.4310/arkiv.2023.v61.n2.a8
Yuta Suzuki
{"title":"On the sum of a prime power and a power in short intervals","authors":"Yuta Suzuki","doi":"10.4310/arkiv.2023.v61.n2.a8","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n2.a8","url":null,"abstract":"Let $R_{k,ell}(N)$ be the representation function for the sum of the $k$-th power of a prime and the $ell$-th power of a positive integer. Languasco and Zaccagnini (2017) proved an asymptotic formula for the average of $R_{1,2}(N)$ over short intervals $(X,X+H]$ of the length $H$ slightly shorter than $X^{frac{1}{2}}$, which is shorter than the length $H=X^{frac{1}{2}+epsilon}$ in the exceptional set estimates of Mikawa (1993) and of Perelli and Pintz (1995). In this paper, we prove that the same asymptotic formula for $R_{1,2}(N)$ holds for $H$ of the size $X^{0.337}$. Recently, Languasco and Zaccagnini (2018) extended their result to more general $(k,ell)$. We also consider this general case, and as a corollary, we prove a conditional result of Languasco and Zaccagnini (2018) for the case $ell=2$ unconditionally up to some small factors.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135783630","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}
引用次数: 0
Two-dimensional Weyl sums failing square-root cancellation along lines 二维Weyl和不能沿直线消去平方根
4区 数学
Arkiv for Matematik Pub Date : 2023-01-01 DOI: 10.4310/arkiv.2023.v61.n2.a1
Julia Brandes, Igor E. Shparlinski
{"title":"Two-dimensional Weyl sums failing square-root cancellation along lines","authors":"Julia Brandes, Igor E. Shparlinski","doi":"10.4310/arkiv.2023.v61.n2.a1","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n2.a1","url":null,"abstract":"We show that a certain two-dimensional family of Weyl sums of length $P$ takes values as large as $P^{3/4 + o(1)}$ on almost all linear slices of the unit torus, contradicting a widely held expectation that Weyl sums should exhibit square-root cancellation on generic subvarieties of the unit torus. This is an extension of a result of J. Brandes, S. T. Parsell, C. Poulias, G. Shakan and R. C. Vaughan (2020) from quadratic and cubic monomials to general polynomials of arbitrary degree. The new ingredients of our approach are the classical results of E. Bombieri (1966) on exponential sums along a curve and R. J. Duffin and A. C. Schaeffer (1941) on Diophantine approximations by rational numbers with prime denominators.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784453","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}
引用次数: 1
Hilbert schemes of points on smooth projective surfaces and generalized Kummer varieties with finite group actions 光滑投影曲面上点的Hilbert格式及有限群作用下的广义Kummer变分
4区 数学
Arkiv for Matematik Pub Date : 2023-01-01 DOI: 10.4310/arkiv.2023.v61.n2.a9
Sailun Zhan
{"title":"Hilbert schemes of points on smooth projective surfaces and generalized Kummer varieties with finite group actions","authors":"Sailun Zhan","doi":"10.4310/arkiv.2023.v61.n2.a9","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n2.a9","url":null,"abstract":"G\"ottsche and Soergel gave formulas for the Hodge numbers of Hilbert schemes of points on a smooth algebraic surface and the Hodge numbers of generalized Kummer varieties. When a smooth projective surface $S$ admits an action by a finite group $G$, we describe the action of $G$ on the Hodge pieces via point counting. Each element of $G$ gives a trace on $sum_{n=0}^{infty}sum_{i=0}^{infty}(-1)^{i}H^{i}(S^{[n]},mathbb{C})q^{n}$. In the case that $S$ is a K3 surface or an abelian surface, the resulting generating functions give some interesting modular forms when $G$ acts faithfully and symplectically on $S$.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135659923","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}
引用次数: 0
A Poisson basis theorem for symmetric algebras of infinite-dimensional Lie algebras 无限维李代数对称代数的泊松基定理
4区 数学
Arkiv for Matematik Pub Date : 2023-01-01 DOI: 10.4310/arkiv.2023.v61.n2.a6
Omar Leon Sanchez, Susan J. Sierra
{"title":"A Poisson basis theorem for symmetric algebras of infinite-dimensional Lie algebras","authors":"Omar Leon Sanchez, Susan J. Sierra","doi":"10.4310/arkiv.2023.v61.n2.a6","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n2.a6","url":null,"abstract":"We consider when the symmetric algebra of an infinite-dimensional Lie algebra, equipped with the natural Poisson bracket, satisfies the ascending chain condition (ACC) on Poisson ideals. We define a combinatorial condition on a graded Lie algebra which we call Dicksonian because it is related to Dickson's lemma on finite subsets of $mathbb N^k$. Our main result is: ","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135784616","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}
引用次数: 3
The spectral picture of Bergman–Toeplitz operators with harmonic polynomial symbols 具有调和多项式符号的Bergman-Toeplitz算子的谱图
4区 数学
Arkiv for Matematik Pub Date : 2023-01-01 DOI: 10.4310/arkiv.2023.v61.n2.a5
Kunyu Guo, Xianfeng Zhao, Dechao Zheng
{"title":"The spectral picture of Bergman–Toeplitz operators with harmonic polynomial symbols","authors":"Kunyu Guo, Xianfeng Zhao, Dechao Zheng","doi":"10.4310/arkiv.2023.v61.n2.a5","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n2.a5","url":null,"abstract":"In this paper, it is shown that some new phenomenon related to the spectra of Toeplitz operators with bounded harmonic symbols on the Bergman space. On one hand, we prove that the spectrum of the Toeplitz operator with symbol ${bar{z}+p}$ is always connected for every polynomial $p$ with degree less than $3$. On the other hand, we show that for each integer $k$ greater than $2$, there exists a polynomial $p$ of degree $k$ such that the spectrum of the Toeplitz operator with symbol ${bar{z}+p}$ has at least one isolated point but has at most finitely many isolated points. Then these results are applied to obtain a class of non-hyponormal Toeplitz operators with bounded harmonic symbols on the Bergman space for which Weyl's theorem holds.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135612976","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}
引用次数: 4
On local and semi-matching colorings of split graphs 关于分割图的局部和半匹配着色
IF 0.7 4区 数学
Arkiv for Matematik Pub Date : 2023-01-01 DOI: 10.4310/arkiv.2023.v61.n1.a10
Y. Shitov
{"title":"On local and semi-matching colorings of split graphs","authors":"Y. Shitov","doi":"10.4310/arkiv.2023.v61.n1.a10","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n1.a10","url":null,"abstract":"","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70393983","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}
引用次数: 0
Regularity of symbolic powers of square-free monomial ideals 无平方单项式理想符号幂的正则性
IF 0.7 4区 数学
Arkiv for Matematik Pub Date : 2023-01-01 DOI: 10.4310/arkiv.2023.v61.n1.a6
Thi Hien Truong, Nam Trung Tran
{"title":"Regularity of symbolic powers of square-free monomial ideals","authors":"Thi Hien Truong, Nam Trung Tran","doi":"10.4310/arkiv.2023.v61.n1.a6","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n1.a6","url":null,"abstract":"","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70394593","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}
引用次数: 1
A complex-analytic approach to streamline properties of deep-water Stokes waves 深水Stokes波流线特性的复杂解析方法
IF 0.7 4区 数学
Arkiv for Matematik Pub Date : 2023-01-01 DOI: 10.4310/arkiv.2023.v61.n1.a5
O. Constantin
{"title":"A complex-analytic approach to streamline properties of deep-water Stokes waves","authors":"O. Constantin","doi":"10.4310/arkiv.2023.v61.n1.a5","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n1.a5","url":null,"abstract":"","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70394540","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}
引用次数: 0
Overcompleteness of coherent frames for unimodular amenable groups 非模可调群的相干框架的过完备性
4区 数学
Arkiv for Matematik Pub Date : 2023-01-01 DOI: 10.4310/arkiv.2023.v61.n2.a2
{"title":"Overcompleteness of coherent frames for unimodular amenable groups","authors":"","doi":"10.4310/arkiv.2023.v61.n2.a2","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n2.a2","url":null,"abstract":". This paper concerns the overcompleteness of coherent frames for amenable unimodular groups. It is shown that for coherent frames associated with an integrable vector a set of positive Beurling density can be removed yet still leave a frame. The obtained results extend various theorems of [J. Fourier Anal. Appl., 12(3):307-344, 2006] to frames with non-Abelian index sets.","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135659927","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}
引用次数: 0
Minimal-mass blow-up solutions for inhomogeneous nonlinear Schrödinger equations with growing potentials 具有增长势的非齐次非线性Schrödinger方程的最小质量爆破解
4区 数学
Arkiv for Matematik Pub Date : 2023-01-01 DOI: 10.4310/arkiv.2023.v61.n2.a7
Naoki Matsui
{"title":"Minimal-mass blow-up solutions for inhomogeneous nonlinear Schrödinger equations with growing potentials","authors":"Naoki Matsui","doi":"10.4310/arkiv.2023.v61.n2.a7","DOIUrl":"https://doi.org/10.4310/arkiv.2023.v61.n2.a7","url":null,"abstract":"","PeriodicalId":55569,"journal":{"name":"Arkiv for Matematik","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135659367","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}
引用次数: 0
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