In this paper, we apply the notion of cocyclic maps to the category of pairs proposed by Hilton and obtain more general concepts. We discuss the concept of cocyclic morphisms with respect to a morphism and find that it is a dual concept of cyclic morphisms with respect to a morphism and a generalization of the notion of cocyclic morphisms with respect to a map. Moreover, we investigate its basic properties including the preservation of cocyclic properties by morphisms and find conditions for which the set of all homotopy classes of cocyclic morphisms with respect to a morphism will have a group structure.
{"title":"COCYCLIC MORPHISM SETS DEPENDING ON A MORPHISM IN THE CATEGORY OF PAIRS","authors":"Jiyean Kim, Keean Lee","doi":"10.4134/BKMS.B190016","DOIUrl":"https://doi.org/10.4134/BKMS.B190016","url":null,"abstract":"In this paper, we apply the notion of cocyclic maps to the category of pairs proposed by Hilton and obtain more general concepts. We discuss the concept of cocyclic morphisms with respect to a morphism and find that it is a dual concept of cyclic morphisms with respect to a morphism and a generalization of the notion of cocyclic morphisms with respect to a map. Moreover, we investigate its basic properties including the preservation of cocyclic properties by morphisms and find conditions for which the set of all homotopy classes of cocyclic morphisms with respect to a morphism will have a group structure.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"56 1","pages":"1589-1600"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70361200","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":"GRADIENT ESTIMATES OF A NONLINEAR ELLIPTIC EQUATION FOR THE V -LAPLACIAN","authors":"Fanqi Zeng","doi":"10.4134/BKMS.B180639","DOIUrl":"https://doi.org/10.4134/BKMS.B180639","url":null,"abstract":"","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"56 1","pages":"853-865"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70359339","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}
Using generalized graded crossed products, we give necessary and sufficient conditions for a simple algebra over a Henselian valued field (under some hypotheses) to have Kummer subfields. This study generalizes some known works. We also study many properties of generalized graded crossed products and conditions for embedding a graded simple algebra into a matrix algebra of a graded division ring.
{"title":"ON GENERALIZED GRADED CROSSED PRODUCTS AND KUMMER SUBFIELDS OF SIMPLE ALGEBRAS","authors":"D. Bennis, Karim Mounirh, Fouad Taraza","doi":"10.4134/BKMS.B180722","DOIUrl":"https://doi.org/10.4134/BKMS.B180722","url":null,"abstract":"Using generalized graded crossed products, we give necessary and sufficient conditions for a simple algebra over a Henselian valued field (under some hypotheses) to have Kummer subfields. This study generalizes some known works. We also study many properties of generalized graded crossed products and conditions for embedding a graded simple algebra into a matrix algebra of a graded division ring.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"77 1","pages":"939-959"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70359520","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}
Da Woon Jung, C. Lee, Yang Lee, Sangwon Park, S. Ryu, Hyo Jin Sung, S. Yun
This article concerns a ring property which preserves the reversibility of elements at nonzero idempotents. A ring R shall be said to be quasi-reversible if 0 6= ab ∈ I(R) for a, b ∈ R implies ba ∈ I(R), where I(R) is the set of all idempotents in R. We investigate the quasireversibility of 2 by 2 full and upper triangular matrix rings over various kinds of reversible rings, concluding that the quasi-reversibility is a proper generalization of the reversibility. It is shown that the quasi-reversibility does not pass to polynomial rings. The structure of Abelian rings is also observed in relation with reversibility and quasi-reversibility. 1. Quasi-reversible rings Throughout every ring is an associative ring with identity unless otherwise stated. Let R be a ring. Use I(R), N∗(R), N(R), and J(R) to denote the set of all idempotents, the upper nilradical (i.e., the sum of all nil ideals), the set of all nilpotent elements, and the Jacobson radical in R, respectively. Note N∗(R) ⊆ N(R). Write I(R)′ = {e ∈ I(R) | e 6= 0}. Z(R) denotes the center of R. The polynomial ring with an indeterminate x over R is denoted by R[x]. Z and Zn denote the ring of integers and the ring of integers modulo n, respectively. Let n ≥ 2. Denote the n by n full (resp., upper triangular) matrix ring over R by Matn(R) (resp., Tn(R)), and Dn(R) = {(aij) ∈ Tn(R) | a11 = · · · = ann}. Use Eij for the matrix with (i, j)-entry 1 and zeros elsewhere, and In denotes the identity matrix in Matn(R). Following Cohn [4], a ring R (possibly without identity) is called reversible if ab = 0 for a, b ∈ R implies ba = 0. Anderson and Camillo [1] used the term ZC2 for the reversibility. A ring (possibly without identity) is usually said to be reduced if it has no nonzero nilpotent elements. Many commutative rings are not reduced (e.g., Znl for n, l ≥ 2), and there exist many noncommutative reduced rings (e.g., direct products of noncommutative domains). It is easily checked that the class of reversible rings contains commutative rings Received August 14, 2018; Revised November 6, 2018; Accepted November 21, 2018. 2010 Mathematics Subject Classification. 16U80, 16S36, 16S50.
{"title":"ON REVERSIBILITY RELATED TO IDEMPOTENTS","authors":"Da Woon Jung, C. Lee, Yang Lee, Sangwon Park, S. Ryu, Hyo Jin Sung, S. Yun","doi":"10.4134/BKMS.B180759","DOIUrl":"https://doi.org/10.4134/BKMS.B180759","url":null,"abstract":"This article concerns a ring property which preserves the reversibility of elements at nonzero idempotents. A ring R shall be said to be quasi-reversible if 0 6= ab ∈ I(R) for a, b ∈ R implies ba ∈ I(R), where I(R) is the set of all idempotents in R. We investigate the quasireversibility of 2 by 2 full and upper triangular matrix rings over various kinds of reversible rings, concluding that the quasi-reversibility is a proper generalization of the reversibility. It is shown that the quasi-reversibility does not pass to polynomial rings. The structure of Abelian rings is also observed in relation with reversibility and quasi-reversibility. 1. Quasi-reversible rings Throughout every ring is an associative ring with identity unless otherwise stated. Let R be a ring. Use I(R), N∗(R), N(R), and J(R) to denote the set of all idempotents, the upper nilradical (i.e., the sum of all nil ideals), the set of all nilpotent elements, and the Jacobson radical in R, respectively. Note N∗(R) ⊆ N(R). Write I(R)′ = {e ∈ I(R) | e 6= 0}. Z(R) denotes the center of R. The polynomial ring with an indeterminate x over R is denoted by R[x]. Z and Zn denote the ring of integers and the ring of integers modulo n, respectively. Let n ≥ 2. Denote the n by n full (resp., upper triangular) matrix ring over R by Matn(R) (resp., Tn(R)), and Dn(R) = {(aij) ∈ Tn(R) | a11 = · · · = ann}. Use Eij for the matrix with (i, j)-entry 1 and zeros elsewhere, and In denotes the identity matrix in Matn(R). Following Cohn [4], a ring R (possibly without identity) is called reversible if ab = 0 for a, b ∈ R implies ba = 0. Anderson and Camillo [1] used the term ZC2 for the reversibility. A ring (possibly without identity) is usually said to be reduced if it has no nonzero nilpotent elements. Many commutative rings are not reduced (e.g., Znl for n, l ≥ 2), and there exist many noncommutative reduced rings (e.g., direct products of noncommutative domains). It is easily checked that the class of reversible rings contains commutative rings Received August 14, 2018; Revised November 6, 2018; Accepted November 21, 2018. 2010 Mathematics Subject Classification. 16U80, 16S36, 16S50.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"56 1","pages":"993-1006"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70360141","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 work, we investigate the following fractional boundary value problems tD T ( |0D t (u(t))|0Dt u(t) ) = ∇W (t, u(t)) + λg(t)|u(t)|q−2u(t), t ∈ (0, T ), u(0) = u(T ) = 0, where ∇W (t, u) is the gradient of W (t, u) at u and W ∈ C([0, T ]×Rn,R) is homogeneous of degree r, λ is a positive parameter, g ∈ C([0, T ]), 1 < r < p < q and 1 p < α < 1. Using the Fibering map and Nehari manifold, for some positive constant λ0 such that 0 < λ < λ0, we prove the existence of at least two non-trivial solutions.
在这项工作中,我们调查以下部分边值问题tD T (| 0 d T (u (T)) | 0 dt u (T)) =∇W (T, u (T)) +λg (T) | u (T) | q−2 u (T) T∈(0,T), u (0) = (T) = 0,∇W (T, u)在哪里的梯度W (T, u) u C和W∈([0,T]×Rn, R)是均匀程度的R,λ是一个积极的参数,g∈C ([0, T]), 1 < R < p < q和p <α< 1。利用纤维映射和Nehari流形，对于某正常数λ0使得0 < λ < λ0，证明了至少两个非平凡解的存在性。
{"title":"NEHARI MANIFOLD AND MULTIPLICITY RESULTS FOR A CLASS OF FRACTIONAL BOUNDARY VALUE PROBLEMS WITH p-LAPLACIAN","authors":"A. Ghanmi, Ziheng Zhang","doi":"10.4134/BKMS.b181172","DOIUrl":"https://doi.org/10.4134/BKMS.b181172","url":null,"abstract":"In this work, we investigate the following fractional boundary value problems tD T ( |0D t (u(t))|0Dt u(t) ) = ∇W (t, u(t)) + λg(t)|u(t)|q−2u(t), t ∈ (0, T ), u(0) = u(T ) = 0, where ∇W (t, u) is the gradient of W (t, u) at u and W ∈ C([0, T ]×Rn,R) is homogeneous of degree r, λ is a positive parameter, g ∈ C([0, T ]), 1 < r < p < q and 1 p < α < 1. Using the Fibering map and Nehari manifold, for some positive constant λ0 such that 0 < λ < λ0, we prove the existence of at least two non-trivial solutions.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"56 1","pages":"1297-1314"},"PeriodicalIF":0.5,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70360893","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, the projectivity of finitely generated flat modules of a commutative ring are studied from a topological point of view. Then various interesting results are obtained. For instance, it is shown that if a ring has either finitely many minimal primes or finitely many maximal ideals then every finitely generated flat module over it is projective. It is also shown that if a particular subset of the prime spectrum of a ring satisfies some certain ascending or descending chain conditions, then every finitely generated flat module over this ring is projective. These results generalize some major results in the literature on the projectivity of finitely generated flat modules.
{"title":"NOTES ON FINITELY GENERATED FLAT MODULES","authors":"A. Tarizadeh","doi":"10.4134/BKMS.B190294","DOIUrl":"https://doi.org/10.4134/BKMS.B190294","url":null,"abstract":"In this paper, the projectivity of finitely generated flat modules of a commutative ring are studied from a topological point of view. Then various interesting results are obtained. For instance, it is shown that if a ring has either finitely many minimal primes or finitely many maximal ideals then every finitely generated flat module over it is projective. It is also shown that if a particular subset of the prime spectrum of a ring satisfies some certain ascending or descending chain conditions, then every finitely generated flat module over this ring is projective. These results generalize some major results in the literature on the projectivity of finitely generated flat modules.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"57 1","pages":"419-427"},"PeriodicalIF":0.5,"publicationDate":"2018-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43236839","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":"A SUFFICIENT CONDITION FOR ACYCLIC 5-CHOOSABILITY OF PLANAR GRAPHS WITHOUT 5-CYCLES","authors":"Lin Sun","doi":"10.4134/BKMS.B170053","DOIUrl":"https://doi.org/10.4134/BKMS.B170053","url":null,"abstract":"","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"55 1","pages":"415-430"},"PeriodicalIF":0.5,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70356850","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 R be an integral domain. We prove that the power series ring R[[X]] is a Krull domain if and only if R[[X]] is a generalized Krull domain and t-dimR ≤ 1, which improves a well-known result of Paran and Temkin. As a consequence we show that one of the following statements holds: (1) the concepts “Krull domain” and “generalized Krull domain” are the same in power series rings, (2) there exists a non-t-SFT domain R with t-dimR > 1 such that t-dimR[[X]] = 1.
{"title":"ON GENERALIZED KRULL POWER SERIES RINGS","authors":"T. Le, T. Phan","doi":"10.4134/BKMS.b170233","DOIUrl":"https://doi.org/10.4134/BKMS.b170233","url":null,"abstract":"Let R be an integral domain. We prove that the power series ring R[[X]] is a Krull domain if and only if R[[X]] is a generalized Krull domain and t-dimR ≤ 1, which improves a well-known result of Paran and Temkin. As a consequence we show that one of the following statements holds: (1) the concepts “Krull domain” and “generalized Krull domain” are the same in power series rings, (2) there exists a non-t-SFT domain R with t-dimR > 1 such that t-dimR[[X]] = 1.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"55 1","pages":"1007-1012"},"PeriodicalIF":0.5,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70358112","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 derive discrete time model of the geometric fractional Brownian motion. It provides numerical pricing scheme of financial derivatives when the market is driven by geometric fractional Brownian motion. With the convergence analysis, we guarantee the convergence of Monte Carlo simulations. The strong convergence rate of our scheme has order H which is Hurst parameter. To obtain our model we need to convert Wick product term of stochastic differential equation into Wick free discrete equation through Malliavin calculus but ours does not include Malliavin derivative term. Finally, we include several numerical experiments for the option pricing.
{"title":"Generating sample paths and their convergence of the geometric fractional brownian motion","authors":"H. Choe, J. Chu, Jong-Eun Kim","doi":"10.4134/BKMS.B170719","DOIUrl":"https://doi.org/10.4134/BKMS.B170719","url":null,"abstract":"We derive discrete time model of the geometric fractional Brownian motion. It provides numerical pricing scheme of financial derivatives when the market is driven by geometric fractional Brownian motion. With the convergence analysis, we guarantee the convergence of Monte Carlo simulations. The strong convergence rate of our scheme has order H which is Hurst parameter. To obtain our model we need to convert Wick product term of stochastic differential equation into Wick free discrete equation through Malliavin calculus but ours does not include Malliavin derivative term. Finally, we include several numerical experiments for the option pricing.","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"55 1","pages":"1241-1261"},"PeriodicalIF":0.5,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70358893","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":"GENERIC LIGHTLIKE SUBMANIFOLDS OF AN INDEFINITE KAEHLER MANIFOLD WITH A QUARTER-SYMMETRIC METRIC CONNECTION","authors":"D. Jin","doi":"10.4134/BKMS.B170093","DOIUrl":"https://doi.org/10.4134/BKMS.B170093","url":null,"abstract":"","PeriodicalId":55301,"journal":{"name":"Bulletin of the Korean Mathematical Society","volume":"73 1","pages":"515-531"},"PeriodicalIF":0.5,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70357399","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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