In this paper we give an alternative proof of the Ohsawa-Takegoshi extension theorem. We prove the theorem using three weight functions which Hörmander used to obtain L estimates for solutions of the ∂̄ problem in pseudoconvex domains.
{"title":"An elementary proof of the Ohsawa-Takegoshi extension theorem","authors":"Adachi Kenzô","doi":"10.5036/MJIU.45.33","DOIUrl":"https://doi.org/10.5036/MJIU.45.33","url":null,"abstract":"In this paper we give an alternative proof of the Ohsawa-Takegoshi extension theorem. We prove the theorem using three weight functions which Hörmander used to obtain L estimates for solutions of the ∂̄ problem in pseudoconvex domains.","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"45 1","pages":"33-51"},"PeriodicalIF":0.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78889823","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Bateman’s transformation is associated with the Lorentzian metric and preserves solutions of the wave equation. We generalize Bateman’s transformation for general indefinite semi-euclidean metrics. Then we show that the generalized transformation preserves solutions of the equation associated with given indefinite metric.
{"title":"Generalizations of Bateman's transformation for general indefinite metrics","authors":"Katsunori Shimomura","doi":"10.5036/MJIU.45.7","DOIUrl":"https://doi.org/10.5036/MJIU.45.7","url":null,"abstract":"Bateman’s transformation is associated with the Lorentzian metric and preserves solutions of the wave equation. We generalize Bateman’s transformation for general indefinite semi-euclidean metrics. Then we show that the generalized transformation preserves solutions of the equation associated with given indefinite metric.","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"22 1","pages":"7-13"},"PeriodicalIF":0.0,"publicationDate":"2013-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81257722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let Ω be a bounded domain of R n . We shall deal with boundary value problems of the following form . (0.1) Here α > 1 − n , u is the relevant solution, ∇ u is its gradient and H is a given real-valued function. Under proper assumptions a pri-ori estimates of solutions u to the problem (0.1) are established by virtue of weighted rearrangement of functions and weighted
{"title":"On the weighted rearrangement of functions and degenerate nonlinear elliptic equations","authors":"Hiroshi Ando, T. Horiuchi","doi":"10.5036/MJIU.44.17","DOIUrl":"https://doi.org/10.5036/MJIU.44.17","url":null,"abstract":"Let Ω be a bounded domain of R n . We shall deal with boundary value problems of the following form . (0.1) Here α > 1 − n , u is the relevant solution, ∇ u is its gradient and H is a given real-valued function. Under proper assumptions a pri-ori estimates of solutions u to the problem (0.1) are established by virtue of weighted rearrangement of functions and weighted","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"29 1","pages":"17-31"},"PeriodicalIF":0.0,"publicationDate":"2012-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80489774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In our previous paper [1], we proved that every transformation which preserves the wave equation is a similarity or a Lorentzian inversion composed with similarities or a Bateman transformation composed with similarities. In this paper, we give several relations between Bateman transformation and Lorentzian inversion. We also prove that only Lorentzian inversion or Bateman transformation is enough to generate the set of all transformations which preserve the wave equation.
{"title":"A relation between the Lorentzian inversion and the Bateman transformation","authors":"Katsunori Shimomura","doi":"10.5036/MJIU.44.1","DOIUrl":"https://doi.org/10.5036/MJIU.44.1","url":null,"abstract":"In our previous paper [1], we proved that every transformation which preserves the wave equation is a similarity or a Lorentzian inversion composed with similarities or a Bateman transformation composed with similarities. In this paper, we give several relations between Bateman transformation and Lorentzian inversion. We also prove that only Lorentzian inversion or Bateman transformation is enough to generate the set of all transformations which preserve the wave equation.","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"65 1","pages":"1-5"},"PeriodicalIF":0.0,"publicationDate":"2012-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74078759","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Liouville type theorem associated with the wave equation","authors":"Katsunori Shimomura","doi":"10.5036/MJIU.43.51","DOIUrl":"https://doi.org/10.5036/MJIU.43.51","url":null,"abstract":"","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"282 1","pages":"51-64"},"PeriodicalIF":0.0,"publicationDate":"2011-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76635267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Based on the famous Mori-Nagata Theorem: The integral closure of a noetherian domain is a Krull domain, similar assertion was conjectured for Mori domain as follows: The complete integral closure of a Mori domain is a Krull domain. The conjecture is positive for a noetherian domain, Krull domain, a semi normal Mori domain [6] and Mori domains for which (D : D*) ≠ 0. In general, as M. Roitman has noted [26], the conjecture is not true. In this paper, an attempt is being made, among other things, to prove that the conjecture is true for a one dimensional Mori domain and for a finite dimensional AV- Mori domain. On the other hand, using the idea of conductor ideals, a simplified proof is given that the conjecture is true for semi normal Mori domains with nonzero pseudo radical.
{"title":"Some results on a conjecture regarding Mori domain","authors":"Habte Gebru","doi":"10.5036/MJIU.43.43","DOIUrl":"https://doi.org/10.5036/MJIU.43.43","url":null,"abstract":"Based on the famous Mori-Nagata Theorem: The integral closure of a noetherian domain is a Krull domain, similar assertion was conjectured for Mori domain as follows: The complete integral closure of a Mori domain is a Krull domain. The conjecture is positive for a noetherian domain, Krull domain, a semi normal Mori domain [6] and Mori domains for which (D : D*) ≠ 0. In general, as M. Roitman has noted [26], the conjecture is not true. In this paper, an attempt is being made, among other things, to prove that the conjecture is true for a one dimensional Mori domain and for a finite dimensional AV- Mori domain. On the other hand, using the idea of conductor ideals, a simplified proof is given that the conjecture is true for semi normal Mori domains with nonzero pseudo radical.","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"6 1","pages":"43-50"},"PeriodicalIF":0.0,"publicationDate":"2011-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74645672","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper generalizes and improves the result of [8] to caloric morphisms between manifolds with different radial semi-euclidean metrics. It is based on the similar arguments as were used in [7] and [8] (cf. [4], [5], [6]), but it succeed to remove the technical assumption from the main result of [8].
{"title":"Caloric morphisms between different radial metrics on semi-euclidean spaces of same dimension","authors":"Katsunori Shimomura","doi":"10.5036/MJIU.43.13","DOIUrl":"https://doi.org/10.5036/MJIU.43.13","url":null,"abstract":"This paper generalizes and improves the result of [8] to caloric morphisms between manifolds with different radial semi-euclidean metrics. It is based on the similar arguments as were used in [7] and [8] (cf. [4], [5], [6]), but it succeed to remove the technical assumption from the main result of [8].","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"27 5 1","pages":"13-41"},"PeriodicalIF":0.0,"publicationDate":"2011-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88889337","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let D be a 1-dimensional Prufer domain with exactly two maximal ideals. We determine the semistar operations on D. Let D be an integral domain, let K be its quotient field, let F(D) be the set of nonzero fractional ideals of D, and let ¯
{"title":"The semistar operations on certain Prüfer domain","authors":"Ryuki Matsuda","doi":"10.5036/MJIU.43.1","DOIUrl":"https://doi.org/10.5036/MJIU.43.1","url":null,"abstract":"Let D be a 1-dimensional Prufer domain with exactly two maximal ideals. We determine the semistar operations on D. Let D be an integral domain, let K be its quotient field, let F(D) be the set of nonzero fractional ideals of D, and let ¯","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"51 1","pages":"1-12"},"PeriodicalIF":0.0,"publicationDate":"2011-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73393719","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A lower bound of nilpotency class of the group of self-homotopy classes","authors":"H. Ōshima","doi":"10.5036/MJIU.42.1","DOIUrl":"https://doi.org/10.5036/MJIU.42.1","url":null,"abstract":"","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"26 1","pages":"1-2"},"PeriodicalIF":0.0,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81994462","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study almost pseudo-valuation semigroups S, especially will study semistar operations on S, and will determine the complete integral closure of S. We will study various cancellation properties of semistar operations on g-monoids. Also, we will study Kronecker function rings of any semistar operations on gmonoids. A. Badawi and E. Houston [BH] introduced an almost pseudo-valuation domain. An integral domain D with quotient field K is called an almost pseudo-valuation domain (or, an APVD) if every prime ideal P of D is strongly primary, that is, if, for elements x, y ∈ K, xy ∈ P and x 6∈ P implies y n ∈ P for some positive integer n. In this paper we will introduce an almost pseudo-valuation semigroup (or, an APVS), and will study it, especially will study semistar operations on an APVS, and will determine the complete integral closure of an APVS. Let G be a torsion-free abelian additive group. A subsemigroup S of G which contains 0 is called a grading monoid (or, a g-monoid). We may confer [M3] for g-monoids. Also, we will study various cancellation properties of semistar operations on g-monoids. Moreover, we will study Kronecker function rings of any semistar operations on g-monoids. The paper consists of seven sections. In §1, we will introduce an APVS, and will show that [BH] holds for g-monoids. In §2, we will show a semigroup version of [KMOS], and will determine the complete integral closure of the APVS. In §3, we will give conditions for an APVS to have only a finite number of semistar operations. In §4, we will study conditions for an APVD to have only a finite number of semistar operations. In §5, we will introduce various cancellation properties of semistar operations on a g-monoid, and will show various implications of the cancellation properties. In §6, we will study results for Kronecker function rings of e.a.b. semistar operations for any semistar operations on g-monoids. §7 is an appendix. Many parts in every §1 ∼ §4 are restatements of [M7]. Since it seems that [M7] has not appeared about six years, and we refered [M7] in
{"title":"Note on g-monoids","authors":"Ryuki Matsuda","doi":"10.5036/MJIU.42.17","DOIUrl":"https://doi.org/10.5036/MJIU.42.17","url":null,"abstract":"We study almost pseudo-valuation semigroups S, especially will study semistar operations on S, and will determine the complete integral closure of S. We will study various cancellation properties of semistar operations on g-monoids. Also, we will study Kronecker function rings of any semistar operations on gmonoids. A. Badawi and E. Houston [BH] introduced an almost pseudo-valuation domain. An integral domain D with quotient field K is called an almost pseudo-valuation domain (or, an APVD) if every prime ideal P of D is strongly primary, that is, if, for elements x, y ∈ K, xy ∈ P and x 6∈ P implies y n ∈ P for some positive integer n. In this paper we will introduce an almost pseudo-valuation semigroup (or, an APVS), and will study it, especially will study semistar operations on an APVS, and will determine the complete integral closure of an APVS. Let G be a torsion-free abelian additive group. A subsemigroup S of G which contains 0 is called a grading monoid (or, a g-monoid). We may confer [M3] for g-monoids. Also, we will study various cancellation properties of semistar operations on g-monoids. Moreover, we will study Kronecker function rings of any semistar operations on g-monoids. The paper consists of seven sections. In §1, we will introduce an APVS, and will show that [BH] holds for g-monoids. In §2, we will show a semigroup version of [KMOS], and will determine the complete integral closure of the APVS. In §3, we will give conditions for an APVS to have only a finite number of semistar operations. In §4, we will study conditions for an APVD to have only a finite number of semistar operations. In §5, we will introduce various cancellation properties of semistar operations on a g-monoid, and will show various implications of the cancellation properties. In §6, we will study results for Kronecker function rings of e.a.b. semistar operations for any semistar operations on g-monoids. §7 is an appendix. Many parts in every §1 ∼ §4 are restatements of [M7]. Since it seems that [M7] has not appeared about six years, and we refered [M7] in","PeriodicalId":18362,"journal":{"name":"Mathematical Journal of Ibaraki University","volume":"78 1","pages":"17-41"},"PeriodicalIF":0.0,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85956134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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