Let R be a G-graded commutative ring with a nonzero unity and P be a proper graded ideal of R. Then P is said to be a graded uniformly pr-ideal of R if there exists n ∈ N such that whenever a, b ∈ h(R) with ab ∈ P and Ann(a) = {0}, then bn ∈ P . The smallest such n is called the order of P and is denoted by ordR(P ). In this article, we study the characterizations on this new class of graded ideals, and investigate the behaviour of graded uniformly pr-ideals in graded factor rings and in direct product of graded rings.
{"title":"GRADED UNIFORMLY pr-IDEALS","authors":"R. Abu-Dawwas, M. Refai","doi":"10.4134/BKMS.B200199","DOIUrl":"https://doi.org/10.4134/BKMS.B200199","url":null,"abstract":"Let R be a G-graded commutative ring with a nonzero unity and P be a proper graded ideal of R. Then P is said to be a graded uniformly pr-ideal of R if there exists n ∈ N such that whenever a, b ∈ h(R) with ab ∈ P and Ann(a) = {0}, then bn ∈ P . The smallest such n is called the order of P and is denoted by ordR(P ). In this article, we study the characterizations on this new class of graded ideals, and investigate the behaviour of graded uniformly pr-ideals in graded factor rings and in direct product of graded rings.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"58 1","pages":"195-204"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70363240","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}
{"title":"CURVATURE ESTIMATES FOR GRADIENT EXPANDING RICCI SOLITONS","authors":"Liangdi Zhang","doi":"10.4134/BKMS.B190805","DOIUrl":"https://doi.org/10.4134/BKMS.B190805","url":null,"abstract":"","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"58 1","pages":"537-557"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70362964","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 use a kind of modified Hurwitz class numbers to express the number of representing a positive integer by some ternary quadratic forms such as the sum of three squares, and the Fourier coefficients of weight 2 modular forms of prime level with trivial character.
{"title":"FOURIER COEFFICIENTS OF WEIGHT 2 MODULAR FORMS OF PRIME LEVEL","authors":"Yan-Bin Li","doi":"10.4134/BKMS.B200376","DOIUrl":"https://doi.org/10.4134/BKMS.B200376","url":null,"abstract":"We use a kind of modified Hurwitz class numbers to express the number of representing a positive integer by some ternary quadratic forms such as the sum of three squares, and the Fourier coefficients of weight 2 modular forms of prime level with trivial character.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"58 1","pages":"593-602"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70363013","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 investigate the properties of Bergman spaces, Bloch spaces and integral means of p-harmonic functions on the unit ball in Rn. Firstly, we offer some Lipschitz-type and double integral characterizations for Bergman space Aγ . Secondly, we characterize Bloch space Bα ω in terms of weighted Lipschitz conditions and BMO functions. Finally, a Hardy-Littlewood type theorem for integral means of p-harmonic functions is established.
{"title":"BERGMAN SPACES, BLOCH SPACES AND INTEGRAL MEANS OF p-HARMONIC FUNCTIONS","authors":"Xi Fu, J. Qiao","doi":"10.4134/BKMS.B200367","DOIUrl":"https://doi.org/10.4134/BKMS.B200367","url":null,"abstract":"In this paper, we investigate the properties of Bergman spaces, Bloch spaces and integral means of p-harmonic functions on the unit ball in Rn. Firstly, we offer some Lipschitz-type and double integral characterizations for Bergman space Aγ . Secondly, we characterize Bloch space Bα ω in terms of weighted Lipschitz conditions and BMO functions. Finally, a Hardy-Littlewood type theorem for integral means of p-harmonic functions is established.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"58 1","pages":"481-495"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70363367","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}
The aim of this paper is to prove the existence of an admissible inertial manifold for mild solutions to infinite delay evolution equation of the form du dt +Au = F (t, ut), t ≥ s, us(θ) = φ(θ), ∀θ ∈ (−∞, 0], s ∈ R, where A is positive definite and self-adjoint with a discrete spectrum, the Lipschitz coefficient of the nonlinear part F may depend on time and belongs to some admissible function space defined on the whole line. The proof is based on the Lyapunov-Perron equation in combination with admissibility and duality estimates.
本文的目的是为了证明一个容许惯性流形的存在轻微解决无限延迟演化方程的形式du / dt +非盟= F (t, ut), t≥年代,我们(θ)=φ(θ),∀θ∈(−∞,0),s∈R,在正定,自伴的离散谱,李普希茨系数的非线性F可能取决于时间和属于一部分容许函数空间上定义整个线。该证明基于李雅普诺夫-佩龙方程，并结合可容许性和对偶性估计。
{"title":"ADMISSIBLE INERTIAL MANIFOLDS FOR INFINITE DELAY EVOLUTION EQUATIONS","authors":"Le Anh Minh","doi":"10.4134/BKMS.B200462","DOIUrl":"https://doi.org/10.4134/BKMS.B200462","url":null,"abstract":"The aim of this paper is to prove the existence of an admissible inertial manifold for mild solutions to infinite delay evolution equation of the form du dt +Au = F (t, ut), t ≥ s, us(θ) = φ(θ), ∀θ ∈ (−∞, 0], s ∈ R, where A is positive definite and self-adjoint with a discrete spectrum, the Lipschitz coefficient of the nonlinear part F may depend on time and belongs to some admissible function space defined on the whole line. The proof is based on the Lyapunov-Perron equation in combination with admissibility and duality estimates.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"58 1","pages":"669-688"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70363580","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 prove global existence and uniqueness of axisymmetric strong solutions for the three dimensional electro-hydrodynamic model based on the coupled Navier–Stokes–Poisson–Nernst–Planck system in the exterior of a cylinder. The key ingredient is that we use the axisymmetry of functions to derive the Lp interpolation inequalities, which allows us to establish all kinds of a priori estimates for the velocity field and charged particles via several cancellation laws.
{"title":"GLOBAL AXISYMMETRIC SOLUTIONS TO THE 3D NAVIER-STOKES-POISSON-NERNST-PLANCK SYSTEM IN THE EXTERIOR OF A CYLINDER","authors":"Jihong Zhao","doi":"10.4134/BKMS.B200536","DOIUrl":"https://doi.org/10.4134/BKMS.B200536","url":null,"abstract":"In this paper we prove global existence and uniqueness of axisymmetric strong solutions for the three dimensional electro-hydrodynamic model based on the coupled Navier–Stokes–Poisson–Nernst–Planck system in the exterior of a cylinder. The key ingredient is that we use the axisymmetry of functions to derive the Lp interpolation inequalities, which allows us to establish all kinds of a priori estimates for the velocity field and charged particles via several cancellation laws.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"58 1","pages":"729-744"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70363833","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 Ω be a proper open subset of Rn and p(·) : Ω → (0, ∞) be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the author introduces the “geometrical” variable Hardy spaces H p(·) r (Ω) and H p(·) z (Ω) on Ω, and then obtains the grand maximal function characterizations of H p(·) r (Ω) and H p(·) z (Ω) when Ω is a strongly Lipschitz domain of Rn. Moreover, the author further introduces the “geometrical” variable local Hardy spaces h p(·) r (Ω), and then establishes the atomic characterization of h p(·) r (Ω) when Ω is a bounded Lipschitz domain of Rn.
{"title":"REAL-VARIABLE CHARACTERIZATIONS OF VARIABLE HARDY SPACES ON LIPSCHITZ DOMAINS OF ℝ n","authors":"Xiong Liu","doi":"10.4134/BKMS.B200545","DOIUrl":"https://doi.org/10.4134/BKMS.B200545","url":null,"abstract":"Let Ω be a proper open subset of Rn and p(·) : Ω → (0, ∞) be a variable exponent function satisfying the globally log-Hölder continuous condition. In this article, the author introduces the “geometrical” variable Hardy spaces H p(·) r (Ω) and H p(·) z (Ω) on Ω, and then obtains the grand maximal function characterizations of H p(·) r (Ω) and H p(·) z (Ω) when Ω is a strongly Lipschitz domain of Rn. Moreover, the author further introduces the “geometrical” variable local Hardy spaces h p(·) r (Ω), and then establishes the atomic characterization of h p(·) r (Ω) when Ω is a bounded Lipschitz domain of Rn.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"58 1","pages":"745-765"},"PeriodicalIF":0.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70363883","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 give some uniqueness theorems of nonconstant meromorphic functions of hyper-order less than one sharing partially three or four small periodic functions with their shifts. As an application, some sufficient conditions for periodicity of meromorphic functions are given. Our results improve and extend previous results of W. Lin, X. Lin and A. Wu [11].
{"title":"ON PARTIAL VALUE SHARING RESULTS OF MEROMORPHIC FUNCTIONS WITH THEIR SHIFTS AND ITS APPLICATIONS","authors":"Vangty Noulorvang, D. T. Pham","doi":"10.4134/BKMS.B190483","DOIUrl":"https://doi.org/10.4134/BKMS.B190483","url":null,"abstract":"In this paper, we give some uniqueness theorems of nonconstant meromorphic functions of hyper-order less than one sharing partially three or four small periodic functions with their shifts. As an application, some sufficient conditions for periodicity of meromorphic functions are given. Our results improve and extend previous results of W. Lin, X. Lin and A. Wu [11].","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"57 1","pages":"1083-1094"},"PeriodicalIF":0.5,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44558168","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}
The aim of this paper is to establish Liouville type results for the stationary MHD equations. In particular, we show that the velocity and magnetic field, belonging to some Lorentz spaces, must be zero. Moreover, we also obtain Liouville type theorem for the case of axially symmetric MHD equations. Our results generalize previous works by Schulz [14] and Seregin-Wang [18].
{"title":"Remarks on Liouville type theorems for the 3D stationary MHD equations","authors":"Zhouyu Li, Pan Liu, P. Niu","doi":"10.4134/BKMS.B190828","DOIUrl":"https://doi.org/10.4134/BKMS.B190828","url":null,"abstract":"The aim of this paper is to establish Liouville type results for the stationary MHD equations. In particular, we show that the velocity and magnetic field, belonging to some Lorentz spaces, must be zero. Moreover, we also obtain Liouville type theorem for the case of axially symmetric MHD equations. Our results generalize previous works by Schulz [14] and Seregin-Wang [18].","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"57 1","pages":"1151-1164"},"PeriodicalIF":0.5,"publicationDate":"2020-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48764252","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 [n] = {1, 2, . . . , n} and 2[n] be the set of all subsets of [n]. For a family F ⊆ 2[n], its diversity, denoted by div(F), is defined to be div(F) = min x∈[n] {|F(x)|} , where F(x) = {F ∈ F : x / ∈ F}. Basically, div(F) measures how far F is from a trivial intersecting family, which is called a star. In this paper, we consider a generalization of diversity for t-intersecting family.
令[n] ={1,2，…， n}和2[n]是[n]的所有子集的集合。对于一个族F∈2[n]，其多样性用div(F)表示，定义为div(F) = min x∈[n] {|F(x)|}，其中F(x) = {F∈F: x /∈F}。基本上，div(F)测量的是F到一个平凡相交族的距离，这个族被称为星形。本文考虑了t相交族的分集性的推广。
{"title":"On diversity of certain T-intersecting families","authors":"C. Y. Ku, K. B. Wong","doi":"10.4134/BKMS.B190301","DOIUrl":"https://doi.org/10.4134/BKMS.B190301","url":null,"abstract":"Let [n] = {1, 2, . . . , n} and 2[n] be the set of all subsets of [n]. For a family F ⊆ 2[n], its diversity, denoted by div(F), is defined to be div(F) = min x∈[n] {|F(x)|} , where F(x) = {F ∈ F : x / ∈ F}. Basically, div(F) measures how far F is from a trivial intersecting family, which is called a star. In this paper, we consider a generalization of diversity for t-intersecting family.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"57 1","pages":"815-829"},"PeriodicalIF":0.5,"publicationDate":"2020-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43287369","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}
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