. Under certain rather weak size conditions assumed on the kernels, some weighted norm inequalities for singular integral operators, related maximal operators, maximal truncated singular integral operators and Marcinkiewicz integral operators in nonisotropic setting will be shown. These weighted norm inequalities will enable us to obtain some vector valued inequalities for the above operators.
{"title":"WEIGHTED ESTIMATES FOR CERTAIN ROUGH OPERATORS WITH APPLICATIONS TO VECTOR VALUED INEQUALITIES","authors":"Feng Liu, Qingying Xue","doi":"10.4134/JKMS.J200450","DOIUrl":"https://doi.org/10.4134/JKMS.J200450","url":null,"abstract":". Under certain rather weak size conditions assumed on the kernels, some weighted norm inequalities for singular integral operators, related maximal operators, maximal truncated singular integral operators and Marcinkiewicz integral operators in nonisotropic setting will be shown. These weighted norm inequalities will enable us to obtain some vector valued inequalities for the above operators.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"58 1","pages":"1035-1058"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70514530","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 algebraic structure of ZpZp[u]/ 〈uk〉-cyclic codes, where uk = 0 and p is a prime. A ZpZp[u]/〈u〉-linear code of length (r + s) is an Rk-submodule of Zp × Rs k with respect to a suitable scalar multiplication, where Rk = Zp[u]/〈u〉. Such a code can also be viewed as an Rk-submodule of Zp[x]/〈x−1〉×Rk[x]/〈x−1〉. A new Gray map has been defined on Zp[u]/〈u〉. We have considered two cases for studying the algebraic structure of ZpZp[u]/〈u〉-cyclic codes, and determined the generator polynomials and minimal spanning sets of these codes in both the cases. In the first case, we have considered (r, p) = 1 and (s, p) 6= 1, and in the second case we consider (r, p) = 1 and (s, p) = 1. We have established the MacWilliams identity for complete weight enumerators of ZpZp[u]/〈u〉-linear codes. Examples have been given to construct ZpZp[u]/〈u〉-cyclic codes, through which we get codes over Zp using the Gray map. Some optimal p-ary codes have been obtained in this way. An example has also been given to illustrate the use of MacWilliams identity.
本文研究了ZpZp[u]/ < uk > -循环码的代数结构，其中uk = 0且p为素数。长度为(r + s)的ZpZp[u]/ < u > -线性码是Zp × Rs k关于合适标量乘法的Rk子模，其中Rk = Zp[u]/ < u >。这样的代码也可以看作是Zp[x]/ < x−1 > ×Rk[x]/ < x−1 >的rk子模块。在Zp[u]/ < u >上定义了一个新的灰色地图。考虑了两种情况下ZpZp[u]/ < u > -循环码的代数结构，并确定了这两种情况下这些码的生成多项式和最小生成集。在第一种情况下，我们考虑(r, p) = 1和(s, p) 6= 1，在第二种情况下，我们考虑(r, p) = 1和(s, p) = 1。建立了ZpZp[u]/ < u > -线性码的完全权枚举数的MacWilliams恒等式。给出了构造ZpZp[u]/ < u > -循环码的实例，并利用灰度图得到了Zp上的码。用这种方法得到了一些最优的p元码。文中还举例说明了MacWilliams恒等式的应用。
{"title":"ON ℤ p ℤ p [u]/ k >-CYCLIC CODES AND THEIR WEIGHT ENUMERATORS","authors":"Maheshanand Bhaintwal, Soumak Biswas","doi":"10.4134/JKMS.J190536","DOIUrl":"https://doi.org/10.4134/JKMS.J190536","url":null,"abstract":"In this paper we study the algebraic structure of ZpZp[u]/ 〈uk〉-cyclic codes, where uk = 0 and p is a prime. A ZpZp[u]/〈u〉-linear code of length (r + s) is an Rk-submodule of Zp × Rs k with respect to a suitable scalar multiplication, where Rk = Zp[u]/〈u〉. Such a code can also be viewed as an Rk-submodule of Zp[x]/〈x−1〉×Rk[x]/〈x−1〉. A new Gray map has been defined on Zp[u]/〈u〉. We have considered two cases for studying the algebraic structure of ZpZp[u]/〈u〉-cyclic codes, and determined the generator polynomials and minimal spanning sets of these codes in both the cases. In the first case, we have considered (r, p) = 1 and (s, p) 6= 1, and in the second case we consider (r, p) = 1 and (s, p) = 1. We have established the MacWilliams identity for complete weight enumerators of ZpZp[u]/〈u〉-linear codes. Examples have been given to construct ZpZp[u]/〈u〉-cyclic codes, through which we get codes over Zp using the Gray map. Some optimal p-ary codes have been obtained in this way. An example has also been given to illustrate the use of MacWilliams identity.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"58 1","pages":"571-595"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70511856","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 set D of column vectors of a generator matrix of a linear code is called a defining set of the linear code. In this paper we consider the problem of constructing few-weight (mainly two- or three-weight) linear codes from defining sets. It can be easily seen that we obtain an one- weight code when we take a defining set to be the nonzero codewords of a linear code. Therefore we have to choose a defining set from a non-linear code to obtain two- or three-weight codes, and we face the problem that the constructed code contains many weights. To overcome this difficulty, we employ the linear codes of the following form: Let D be a subset of F n 2 , and W (resp. V ) be a subspace of F 2 (resp. F n 2 ). We define the linear code C D ( W ; V ) with defining set D and restricted to W,V by C D ( W ; V ) = { ( s + u · x ) x ∈ D ∗ | s ∈ W,u ∈ V } . We obtain two- or three-weight codes by taking D to be a Vasil’ev code of length n = 2 m − 1( m ≥ 3) and a suitable choices of W . We do the same job for D being the complement of a Vasil’ev code. The constructed few-weight codes share some nice properties. Some of them are optimal in the sense that they attain either the Griesmer bound or the Grey-Rankin bound. Most of them are minimal codes which, in turn, have an application in secret sharing schemes. Finally we obtain an infinite family of minimal codes for which the sufficient condition of Ashikhmin and Barg does not hold.
。线性码的生成矩阵的列向量的集合D称为线性码的定义集合。本文研究了从定义集构造少权(主要是二权或三权)线性码的问题。很容易看出，当我们取一个定义集为线性码的非零码字时，我们得到一个一权码。因此，我们必须从非线性码中选择一个定义集来获得二权码或三权码，并且我们面临着构造的码包含多个权值的问题。为了克服这个困难，我们采用以下形式的线性编码:设D是fn2的一个子集，W (p = 1)。V)是f2的一个子空间。F n 2)。我们定义线性码C D (W;V)，定义集合D，限定为W,V受C D (W;V) = {(s + u·x) x∈D∗| s∈W,u∈V}。取D为长度为n = 2m - 1(m≥3)的Vasil 'ev码，并选择合适的W，得到两权码或三权码。对于D是Vasil 'ev代码的补充，我们做同样的工作。构造的低权重代码共享一些很好的属性。其中一些是最优的，因为它们达到了Griesmer界或Grey-Rankin界。它们中的大多数都是最小的代码，反过来在秘密共享方案中有应用。最后，我们得到了不成立Ashikhmin和Barg充分条件的无穷极小码族。
{"title":"CONSTRUCTION OF TWO- OR THREE-WEIGHT BINARY LINEAR CODES FROM VASIL'EV CODES","authors":"J. Hyun, Jaeseon Kim","doi":"10.4134/JKMS.J190429","DOIUrl":"https://doi.org/10.4134/JKMS.J190429","url":null,"abstract":". The set D of column vectors of a generator matrix of a linear code is called a defining set of the linear code. In this paper we consider the problem of constructing few-weight (mainly two- or three-weight) linear codes from defining sets. It can be easily seen that we obtain an one- weight code when we take a defining set to be the nonzero codewords of a linear code. Therefore we have to choose a defining set from a non-linear code to obtain two- or three-weight codes, and we face the problem that the constructed code contains many weights. To overcome this difficulty, we employ the linear codes of the following form: Let D be a subset of F n 2 , and W (resp. V ) be a subspace of F 2 (resp. F n 2 ). We define the linear code C D ( W ; V ) with defining set D and restricted to W,V by C D ( W ; V ) = { ( s + u · x ) x ∈ D ∗ | s ∈ W,u ∈ V } . We obtain two- or three-weight codes by taking D to be a Vasil’ev code of length n = 2 m − 1( m ≥ 3) and a suitable choices of W . We do the same job for D being the complement of a Vasil’ev code. The constructed few-weight codes share some nice properties. Some of them are optimal in the sense that they attain either the Griesmer bound or the Grey-Rankin bound. Most of them are minimal codes which, in turn, have an application in secret sharing schemes. Finally we obtain an infinite family of minimal codes for which the sufficient condition of Ashikhmin and Barg does not hold.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"58 1","pages":"29-44"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70512066","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 weighted norm inequalities for rough singular integrals along surfaces with radial kernels h and sphere kernels Ω by assuming h ∈ (cid:52) γ ( R + ) and Ω ∈ WG β (S n − 1 ) for some γ > 1 and β > 1. Here Ω ∈ WG β (S n − 1 ) denotes the variant of Grafakos-Stefanov type size conditions on the unit sphere. Our results essentially improve and extend the previous weighted results for the rough singular integrals and the corresponding maximal truncated operators.
。本文通过假设h∈(cid:52) γ (R +)和Ω∈WG β (S n−1)对于某些γ > 1和β > 1，证明了沿径向核为h和球核为Ω的粗糙奇异积分的加权范数不等式。其中Ω∈WG β (S n−1)表示单位球上的Grafakos-Stefanov型尺寸条件的变体。我们的结果从本质上改进和推广了粗糙奇异积分和相应的极大截断算子的加权结果。
{"title":"WEIGHTED L p -BOUNDEDNESS OF SINGULAR INTEGRALS WITH ROUGH KERNEL ASSOCIATED TO SURFACES","authors":"Ronghui Liu, Huo-xiong Wu","doi":"10.4134/JKMS.J190845","DOIUrl":"https://doi.org/10.4134/JKMS.J190845","url":null,"abstract":". In this paper, we prove weighted norm inequalities for rough singular integrals along surfaces with radial kernels h and sphere kernels Ω by assuming h ∈ (cid:52) γ ( R + ) and Ω ∈ WG β (S n − 1 ) for some γ > 1 and β > 1. Here Ω ∈ WG β (S n − 1 ) denotes the variant of Grafakos-Stefanov type size conditions on the unit sphere. Our results essentially improve and extend the previous weighted results for the rough singular integrals and the corresponding maximal truncated operators.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"58 1","pages":"69-90"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70513172","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 main study the strong law of large numbers and complete convergence for weighted sums of asymptotically almost negatively associated (AANA, in short) random variables, by using the Marcinkiewicz-Zygmund type moment inequality and Roenthal type moment inequality for AANA random variables. As an application, the complete consistency for the weighted linear estimator of nonparametric regression models based on AANA errors is obtained. Finally, some numerical simulations are carried out to verify the validity of our theoretical result.
{"title":"COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF AANA RANDOM VARIABLES AND ITS APPLICATION IN NONPARAMETRIC REGRESSION MODELS","authors":"A. Shen, Yajing Zhang","doi":"10.4134/JKMS.J200029","DOIUrl":"https://doi.org/10.4134/JKMS.J200029","url":null,"abstract":"In this paper, we main study the strong law of large numbers and complete convergence for weighted sums of asymptotically almost negatively associated (AANA, in short) random variables, by using the Marcinkiewicz-Zygmund type moment inequality and Roenthal type moment inequality for AANA random variables. As an application, the complete consistency for the weighted linear estimator of nonparametric regression models based on AANA errors is obtained. Finally, some numerical simulations are carried out to verify the validity of our theoretical result.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"58 1","pages":"327-349"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70513813","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 the present paper, we are concerned with the existence of ground state sign-changing solutions for the following SchrödingerPoisson-Kirchhoff system − (1+b ∫ R3 |∇u|dx)4u+V (x)u+k(x)φu=λf(x)u+|u|u, in R, −4φ=k(x)u, in R, where b > 0, V (x), k(x) and f(x) are positive continuous smooth functions; 0 < λ < λ1 and λ1 is the first eigenvalue of the problem −4u + V (x)u = λf(x)u in H. With the help of the constraint variational method, we obtain that the Schrödinger-Poisson-Kirchhoff type system possesses at least one ground state sign-changing solution for all b > 0 and 0 < λ < λ1. Moreover, we prove that its energy is strictly larger than twice that of the ground state solutions of Nehari type.
{"title":"GROUND STATE SIGN-CHANGING SOLUTIONS FOR A CLASS OF SCHRÖDINGER-POISSON-KIRCHHOFF TYPEPROBLEMS WITH A CRITICAL NONLINEARITY IN ℝ 3","authors":"Aixia Qian, M. Zhang","doi":"10.4134/JKMS.J200497","DOIUrl":"https://doi.org/10.4134/JKMS.J200497","url":null,"abstract":"In the present paper, we are concerned with the existence of ground state sign-changing solutions for the following SchrödingerPoisson-Kirchhoff system − (1+b ∫ R3 |∇u|dx)4u+V (x)u+k(x)φu=λf(x)u+|u|u, in R, −4φ=k(x)u, in R, where b > 0, V (x), k(x) and f(x) are positive continuous smooth functions; 0 < λ < λ1 and λ1 is the first eigenvalue of the problem −4u + V (x)u = λf(x)u in H. With the help of the constraint variational method, we obtain that the Schrödinger-Poisson-Kirchhoff type system possesses at least one ground state sign-changing solution for all b > 0 and 0 < λ < λ1. Moreover, we prove that its energy is strictly larger than twice that of the ground state solutions of Nehari type.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"58 1","pages":"1181-1209"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70514212","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 is concerned with a reaction-diffusion logistic model. In [17], Lou observed that a heterogeneous environment with diffusion makes the total biomass greater than the total carrying capacity. Regarding the ratio of biomass to carrying capacity, Ni [10] raised a conjecture that the ratio has a upper bound depending only on the spatial dimension. For the one-dimensional case, Bai, He, and Li [1] proved that the optimal upper bound is 3. Recently, Inoue and Kuto [13] showed that the supremum of the ratio is infinity when the domain is a multi-dimensional ball. In this paper, we generalized the result of [13] to an arbitrary smooth bounded domain in Rn, n ≥ 2. We use the subsolution and super-solution method. The idea of the proof is essentially the same as the proof of [13] but we have improved the construction of sub-solutions. This is the complete answer to the conjecture of Ni.
本文研究了一个反应-扩散逻辑模型。在[17]中，Lou观察到具有扩散的异质环境使得总生物量大于总承载能力。对于生物量与承载能力的比值，Ni[10]提出了该比值仅依赖于空间维度有上界的猜想。对于一维情况，Bai, He, and Li[1]证明了最优上界为3。最近，Inoue和Kuto[13]证明了当域是一个多维球时，该比值的上极值为无穷大。本文将[13]的结果推广到Rn, n≥2中的任意光滑有界区域。我们采用了亚解法和超解法。证明的思想本质上与[13]的证明相同，但我们改进了子解的构造。这就是Ni猜想的完整答案。
{"title":"ON THE RATIO OF BIOMASS TO TOTAL CARRYING CAPACITY IN HIGH DIMENSIONS","authors":"Jun-Haeng Heo, Yeonho Kim","doi":"10.4134/JKMS.J200538","DOIUrl":"https://doi.org/10.4134/JKMS.J200538","url":null,"abstract":"This paper is concerned with a reaction-diffusion logistic model. In [17], Lou observed that a heterogeneous environment with diffusion makes the total biomass greater than the total carrying capacity. Regarding the ratio of biomass to carrying capacity, Ni [10] raised a conjecture that the ratio has a upper bound depending only on the spatial dimension. For the one-dimensional case, Bai, He, and Li [1] proved that the optimal upper bound is 3. Recently, Inoue and Kuto [13] showed that the supremum of the ratio is infinity when the domain is a multi-dimensional ball. In this paper, we generalized the result of [13] to an arbitrary smooth bounded domain in Rn, n ≥ 2. We use the subsolution and super-solution method. The idea of the proof is essentially the same as the proof of [13] but we have improved the construction of sub-solutions. This is the complete answer to the conjecture of Ni.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"58 1","pages":"1227-1237"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70514257","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 K be a number field and L a finite abelian extension of K . Let E be an elliptic curve defined over K . The restriction of scalars Res LK E decomposes (up to isogeny) into abelian varieties over K Res LK E ∼ (cid:77) F ∈ S A F , where S is the set of cyclic extensions of K in L . It is known that if L is a quadratic extension, then A L is the quadratic twist of E . In this paper, we consider the case that K is a number field containing a primitive third root of unity, L = K ( 3 √ D ) is the cyclic cubic extension of K for some D ∈ K × / ( K × ) 3 , E = E a : y 2 = x 3 + a is an elliptic curve with j invariant 0 defined over K , and E Da : y 2 = x 3 + aD 2 is the cubic twist of E a . In this case, we prove A L is isogenous over K to E Da × E D 2 a and a property of the Selmer rank of A L , which is a cubic analogue of a theorem of Mazur and Rubin on quadratic twists.
． 设K是一个数字域，L是K的有限阿贝尔扩展。设E是一条定义在K上的椭圆曲线。标量的限制Res LK E分解为K上的阿贝尔变体Res LK E ~ (cid:77) F∈S A F，其中S是K在L中的循环扩展的集合。已知，如果L是二次扩展，则L是E的二次扭转。在这篇文章中,我们考虑的K是一个数字字段包含一个原始的第三根的团结,L = K(√3 D)的循环立方扩展的K D∈K / (K)××3,E = E: 2 y = x 3 + 0是一个j的椭圆曲线不变的定义/ K,和E Da: y = x 3 +广告2是立方扭曲的E。在这种情况下，我们证明了A L在K到E Da × E d2 A上是等齐次的，并证明了A L的Selmer秩的一个性质，它是Mazur和Rubin关于二次旋的定理的三次类似。
{"title":"RESTRICTION OF SCALARS AND CUBIC TWISTS OF ELLIPTIC CURVES","authors":"Dongho Byeon, Keunyoung Jeong, N. Kim","doi":"10.4134/JKMS.J190867","DOIUrl":"https://doi.org/10.4134/JKMS.J190867","url":null,"abstract":". Let K be a number field and L a finite abelian extension of K . Let E be an elliptic curve defined over K . The restriction of scalars Res LK E decomposes (up to isogeny) into abelian varieties over K Res LK E ∼ (cid:77) F ∈ S A F , where S is the set of cyclic extensions of K in L . It is known that if L is a quadratic extension, then A L is the quadratic twist of E . In this paper, we consider the case that K is a number field containing a primitive third root of unity, L = K ( 3 √ D ) is the cyclic cubic extension of K for some D ∈ K × / ( K × ) 3 , E = E a : y 2 = x 3 + a is an elliptic curve with j invariant 0 defined over K , and E Da : y 2 = x 3 + aD 2 is the cubic twist of E a . In this case, we prove A L is isogenous over K to E Da × E D 2 a and a property of the Selmer rank of A L , which is a cubic analogue of a theorem of Mazur and Rubin on quadratic twists.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"58 1","pages":"123-132"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70514011","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 article, we study the Klein-Gordon-Maxwell equations arising from a semilocal gauge field model. This model describes the interaction of two complex scalar fields and one gauge field, and generalizes the classical Klein-Gordon equation coupled with the Maxwell electrodynamics. We prove that there exist infinitely many standing wave solutions for p ∈ (2, 6) which are radially symmetric. Here, p comes from the exponent of the potential of scalar fields. We also prove the nonexistence of nontrivial solutions for the critical case p = 6.
{"title":"ON SEMILOCAL KLEIN-GORDON-MAXWELL EQUATIONS","authors":"Jongmin Han, Juhee Sohn, Yeong Seok Yoo","doi":"10.4134/JKMS.J200463","DOIUrl":"https://doi.org/10.4134/JKMS.J200463","url":null,"abstract":"In this article, we study the Klein-Gordon-Maxwell equations arising from a semilocal gauge field model. This model describes the interaction of two complex scalar fields and one gauge field, and generalizes the classical Klein-Gordon equation coupled with the Maxwell electrodynamics. We prove that there exist infinitely many standing wave solutions for p ∈ (2, 6) which are radially symmetric. Here, p comes from the exponent of the potential of scalar fields. We also prove the nonexistence of nontrivial solutions for the critical case p = 6.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"58 1","pages":"1131-1145"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70514139","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 main focus of the present paper is to present new set of sufficient conditions so that the normalized form of the Mittag-Leffler and Wright functions have certain geometric properties like close-to-convexity, univalency, convexity and starlikeness inside the unit disk. Interesting consequences and examples are derived to support that these results are better than the existing ones and improve several results available in the literature.
{"title":"ON GEOMETRIC PROPERTIES OF THE MITTAG-LEFFLER AND WRIGHT FUNCTIONS","authors":"Sourav Das, K. Mehrez","doi":"10.4134/JKMS.J200333","DOIUrl":"https://doi.org/10.4134/JKMS.J200333","url":null,"abstract":"The main focus of the present paper is to present new set of sufficient conditions so that the normalized form of the Mittag-Leffler and Wright functions have certain geometric properties like close-to-convexity, univalency, convexity and starlikeness inside the unit disk. Interesting consequences and examples are derived to support that these results are better than the existing ones and improve several results available in the literature.","PeriodicalId":49993,"journal":{"name":"Journal of the Korean Mathematical Society","volume":"58 1","pages":"949-965"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70514473","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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