In the present work, the complete convergence and complete moment convergence properties for arrays of rowwise extended negatively dependent (END) random variables are investigated. Some sharp theorems on these strong convergence for weighted sums of END cases are established. These main results not only generalize the known corresponding ones of Cai [2], Wang et al. [17] and Shen [14], but also improve them, respectively.
本文研究了行扩展负相关随机变量数组的完全收敛性和完全矩收敛性。建立了END情况下加权和的强收敛性的一些尖锐定理。这些主要结果不仅对Cai[2]、Wang et al.[17]和Shen[14]的已知对应结果进行了推广，而且对其进行了改进。
{"title":"COMPLETE CONVERGENCE AND COMPLETE MOMENT CONVERGENCE THEOREMS FOR WEIGHTED SUMS OF ARRAYS OF ROWWISE EXTENDED NEGATIVELY DEPENDENT RANDOM VARIABLES","authors":"Haiwu Huang, QingXia Zhang","doi":"10.4134/BKMS.B180791","DOIUrl":"https://doi.org/10.4134/BKMS.B180791","url":null,"abstract":"In the present work, the complete convergence and complete moment convergence properties for arrays of rowwise extended negatively dependent (END) random variables are investigated. Some sharp theorems on these strong convergence for weighted sums of END cases are established. These main results not only generalize the known corresponding ones of Cai [2], Wang et al. [17] and Shen [14], but also improve them, respectively.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"56 1","pages":"1007-1025"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70360202","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 E → M be an arbitrary vector bundle of rank k over a Riemannian manifold M equipped with a fiber metric and a compatible connection DE . R. Albuquerque constructed a general class of (twoweights) spherically symmetric metrics on E. In this paper, we give a characterization of locally symmetric spherically symmetric metrics on E in the case when DE is flat. We study also the Einstein property on E proving, among other results, that if k ≥ 2 and the base manifold is Einstein with positive constant scalar curvature, then there is a 1parameter family of Einstein spherically symmetric metrics on E, which are not Ricci-flat. Introduction and main results In the framework of Riemannian geometry, many special kinds of vector bundles have been considered and extensively studied, such as the cotangent bundle or the tangent bundle the literature of whose is very rich. Indeed, a wide range of interesting works have been published on the geometry of tangent bundles endowed with special types of metrics (Sasaki, Cheeger-Gromoll, . . . ) or more generally with g-natural metrics (cf. [1–3], [7]). For the general case of an arbitrary vector bundle, to the best of our knowledge, the situation becomes substantially different (cf. [5], [6]). Let (E, π,M) be a vector bundle equipped with a fiber metric h and a connection D compatible with h. Classically, the total space E, as a Riemannian manifold, have been “naturally” equipped with the metric π∗g ⊕ πh. More recently, in [4], R. Albuquerque considered a more general class of two-weights metrics with the weight functions depending on the fibre norm of E, i.e., metrics of the form g̃ = e2φ1π∗g ⊕ e2πh, where φ1, φ2 are smooth scalar functions on E depending only of the norm r = h(e, e) for e ∈ E, and smooth at r = 0 on the right. He called such metrics Received October 16, 2018; Revised March 1, 2019; Accepted March 8, 2019. 2010 Mathematics Subject Classification. 53C07, 53C24, 53C25.
{"title":"ON THE GEOMETRY OF VECTOR BUNDLES WITH FLAT CONNECTIONS","authors":"M. Abbassi, Ibrahim Lakrini","doi":"10.4134/BKMS.B180983","DOIUrl":"https://doi.org/10.4134/BKMS.B180983","url":null,"abstract":"Let E → M be an arbitrary vector bundle of rank k over a Riemannian manifold M equipped with a fiber metric and a compatible connection DE . R. Albuquerque constructed a general class of (twoweights) spherically symmetric metrics on E. In this paper, we give a characterization of locally symmetric spherically symmetric metrics on E in the case when DE is flat. We study also the Einstein property on E proving, among other results, that if k ≥ 2 and the base manifold is Einstein with positive constant scalar curvature, then there is a 1parameter family of Einstein spherically symmetric metrics on E, which are not Ricci-flat. Introduction and main results In the framework of Riemannian geometry, many special kinds of vector bundles have been considered and extensively studied, such as the cotangent bundle or the tangent bundle the literature of whose is very rich. Indeed, a wide range of interesting works have been published on the geometry of tangent bundles endowed with special types of metrics (Sasaki, Cheeger-Gromoll, . . . ) or more generally with g-natural metrics (cf. [1–3], [7]). For the general case of an arbitrary vector bundle, to the best of our knowledge, the situation becomes substantially different (cf. [5], [6]). Let (E, π,M) be a vector bundle equipped with a fiber metric h and a connection D compatible with h. Classically, the total space E, as a Riemannian manifold, have been “naturally” equipped with the metric π∗g ⊕ πh. More recently, in [4], R. Albuquerque considered a more general class of two-weights metrics with the weight functions depending on the fibre norm of E, i.e., metrics of the form g̃ = e2φ1π∗g ⊕ e2πh, where φ1, φ2 are smooth scalar functions on E depending only of the norm r = h(e, e) for e ∈ E, and smooth at r = 0 on the right. He called such metrics Received October 16, 2018; Revised March 1, 2019; Accepted March 8, 2019. 2010 Mathematics Subject Classification. 53C07, 53C24, 53C25.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"56 1","pages":"1219-1233"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70360795","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 well known that the denominator of the Dedekind sum s(c, d) divides 2 gcd(d, 3)d and that no smaller denominator independent of c can be expected. In contrast, here we prove that we usually get a smaller denominator in S(H, d), the sum of the s(c, d)’s over all the c’s in a subgroup H of order n > 1 in the multiplicative group (Z/dZ)∗. First, we prove that for p > 3 a prime, the sum 2S(H, p) is a rational integer of the same parity as (p−1)/2. We give an application of this result to upper bounds on relative class numbers of imaginary abelian number fields of prime conductor. Finally, we give a general result on the denominator of S(H, d) for non necessarily prime d’s. We show that its denominator is a divisor of some explicit divisor of 2d gcd(d, 3).
{"title":"ON THE DENOMINATOR OF DEDEKIND SUMS","authors":"S. Louboutin","doi":"10.4134/BKMS.B180043","DOIUrl":"https://doi.org/10.4134/BKMS.B180043","url":null,"abstract":"It is well known that the denominator of the Dedekind sum s(c, d) divides 2 gcd(d, 3)d and that no smaller denominator independent of c can be expected. In contrast, here we prove that we usually get a smaller denominator in S(H, d), the sum of the s(c, d)’s over all the c’s in a subgroup H of order n > 1 in the multiplicative group (Z/dZ)∗. First, we prove that for p > 3 a prime, the sum 2S(H, p) is a rational integer of the same parity as (p−1)/2. We give an application of this result to upper bounds on relative class numbers of imaginary abelian number fields of prime conductor. Finally, we give a general result on the denominator of S(H, d) for non necessarily prime d’s. We show that its denominator is a divisor of some explicit divisor of 2d gcd(d, 3).","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"56 1","pages":"815-827"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70358527","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}
For a simple implementation, a linear convex splitting scheme was coupled with the Fourier spectral method for the Cahn–Hilliard equation with a logarithmic free energy. However, an inappropriate value of the splitting parameter of the linear scheme may lead to incorrect morphologies in the phase separation process. In order to overcome this problem, we present a nonlinear convex splitting Fourier spectral scheme for the Cahn–Hilliard equation with a logarithmic free energy, which is an appropriate extension of Eyre’s idea of convex-concave decomposition of the energy functional. Using the nonlinear scheme, we derive a useful formula for the relation between the gradient energy coefficient and the thickness of the interfacial layer. And we present numerical simulations showing the different evolution of the solution using the linear and nonlinear schemes. The numerical results demonstrate that the nonlinear scheme is more accurate than the linear one.
{"title":"A nonlinear convex splitting fourier spectral scheme for the Cahn–Hilliard equation with a logarithmic free energy","authors":"Junseok Kim, H. Lee","doi":"10.4134/BKMS.B180238","DOIUrl":"https://doi.org/10.4134/BKMS.B180238","url":null,"abstract":"For a simple implementation, a linear convex splitting scheme was coupled with the Fourier spectral method for the Cahn–Hilliard equation with a logarithmic free energy. However, an inappropriate value of the splitting parameter of the linear scheme may lead to incorrect morphologies in the phase separation process. In order to overcome this problem, we present a nonlinear convex splitting Fourier spectral scheme for the Cahn–Hilliard equation with a logarithmic free energy, which is an appropriate extension of Eyre’s idea of convex-concave decomposition of the energy functional. Using the nonlinear scheme, we derive a useful formula for the relation between the gradient energy coefficient and the thickness of the interfacial layer. And we present numerical simulations showing the different evolution of the solution using the linear and nonlinear schemes. The numerical results demonstrate that the nonlinear scheme is more accurate than the linear one.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"56 1","pages":"265-276"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70359317","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 Γ be a nonzero commutative cancellative monoid (written additively), R = ⊕ α∈Γ Rα be a Γ-graded integral domain with Rα 6= {0} for all α ∈ Γ, and S(H) = {f ∈ R |C(f) = R}. In this paper, we study homogeneously divisorial domains which are graded integral domains whose nonzero homogeneous ideals are divisorial. Among other things, we show that if R is integrally closed, then R is a homogeneously divisorial domain if and only if RS(H) is an h-local Prüfer domain whose maximal ideals are invertible, if and only if R satisfies the following four conditions: (i) R is a graded-Prüfer domain, (ii) every homogeneous maximal ideal of R is invertible, (iii) each nonzero homogeneous prime ideal of R is contained in a unique homogeneous maximal ideal, and (iv) each homogeneous ideal of R has only finitely many minimal prime ideals. We also show that if R is a graded-Noetherian domain, then R is a homogeneously divisorial domain if and only if RS(H) is a divisorial domain of (Krull) dimension one.
{"title":"GRADED INTEGRAL DOMAINS IN WHICH EACH NONZERO HOMOGENEOUS IDEAL IS DIVISORIAL","authors":"G. Chang, Haleh Hamdi, P. Sahandi","doi":"10.4134/BKMS.b180870","DOIUrl":"https://doi.org/10.4134/BKMS.b180870","url":null,"abstract":"Let Γ be a nonzero commutative cancellative monoid (written additively), R = ⊕ α∈Γ Rα be a Γ-graded integral domain with Rα 6= {0} for all α ∈ Γ, and S(H) = {f ∈ R |C(f) = R}. In this paper, we study homogeneously divisorial domains which are graded integral domains whose nonzero homogeneous ideals are divisorial. Among other things, we show that if R is integrally closed, then R is a homogeneously divisorial domain if and only if RS(H) is an h-local Prüfer domain whose maximal ideals are invertible, if and only if R satisfies the following four conditions: (i) R is a graded-Prüfer domain, (ii) every homogeneous maximal ideal of R is invertible, (iii) each nonzero homogeneous prime ideal of R is contained in a unique homogeneous maximal ideal, and (iv) each homogeneous ideal of R has only finitely many minimal prime ideals. We also show that if R is a graded-Noetherian domain, then R is a homogeneously divisorial domain if and only if RS(H) is a divisorial domain of (Krull) dimension one.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"56 1","pages":"1041-1057"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70360399","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 establish a monotonicity formula of V -harmonic maps by using the stress-energy tensor. Use the monotonicity formula, we can derive a Liouville type theorem for V -harmonic maps. As applications, we also obtain monotonicity and constancy of Weyl harmonic maps from conformal manifolds to Riemannian manifolds and ±holomorphic maps between almost Hermitian manifolds. Finally, a constant boundary-value problem of V -harmonic maps is considered.
{"title":"A MONOTONICITY FORMULA AND A LIOUVILLE TYPE THEOREM OF V-HARMONIC MAPS","authors":"Guangwen Zhao","doi":"10.4134/BKMS.B181181","DOIUrl":"https://doi.org/10.4134/BKMS.B181181","url":null,"abstract":"We establish a monotonicity formula of V -harmonic maps by using the stress-energy tensor. Use the monotonicity formula, we can derive a Liouville type theorem for V -harmonic maps. As applications, we also obtain monotonicity and constancy of Weyl harmonic maps from conformal manifolds to Riemannian manifolds and ±holomorphic maps between almost Hermitian manifolds. Finally, a constant boundary-value problem of V -harmonic maps is considered.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"56 1","pages":"1327-1340"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70361221","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":"$mathcal{F}_{mathcal{S}}$-MITTAG-LEFFLER MODULES AND GLOBAL DIMENSION RELATIVE TO $mathcal{F}_{mathcal{S}}$-MITTAG-LEFFLER MODULES","authors":"Mingzhao Chen, Fanggui Wang","doi":"10.4134/BKMS.B180740","DOIUrl":"https://doi.org/10.4134/BKMS.B180740","url":null,"abstract":"","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"38 1","pages":"961-976"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70360415","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 consider a semilinear stochastic heat equation driven by an additive fractional white noise. Under the pathwise uniqueness property, we establish various strong stability results. As a consequence, we give an application to the convergence of the Picard successive approximation.
{"title":"SOME STABILITY RESULTS FOR SEMILINEAR STOCHASTIC HEAT EQUATION DRIVEN BY A FRACTIONAL NOISE","authors":"Oussama El Barrimi, Y. Ouknine","doi":"10.4134/BKMS.B180445","DOIUrl":"https://doi.org/10.4134/BKMS.B180445","url":null,"abstract":"In this paper, we consider a semilinear stochastic heat equation driven by an additive fractional white noise. Under the pathwise uniqueness property, we establish various strong stability results. As a consequence, we give an application to the convergence of the Picard successive approximation.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"56 1","pages":"631-648"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70359380","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":"NEGACYCLIC CODES OF LENGTH 8p s OVER F p m + uF p m","authors":"C. Klin-eam, Jirayu Phuto","doi":"10.4134/BKMS.B180721","DOIUrl":"https://doi.org/10.4134/BKMS.B180721","url":null,"abstract":"","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"56 1","pages":"1385-1422"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70359955","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 a theory for the structure of τwLoewy series of modules over commutative rings, where τw is the hereditary torsion theory induced by the so-called w-operation, and explore the relationship between τw-Loewy modules and w-Artinian modules.
{"title":"τ w -LOEWY MODULES AND THEIR APPLICATIONS","authors":"Hwankoo Kim, J. Lim, D. Zhou","doi":"10.4134/BKMS.B190049","DOIUrl":"https://doi.org/10.4134/BKMS.B190049","url":null,"abstract":"In this paper, we study a theory for the structure of τwLoewy series of modules over commutative rings, where τw is the hereditary torsion theory induced by the so-called w-operation, and explore the relationship between τw-Loewy modules and w-Artinian modules.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"56 1","pages":"1617-1642"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70361400","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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