We study parabolic, ridge and sub-parabolic curves on implicit surfaces defined by smooth functions R-equivalent to A1 -singularity. To investigate ridge and sub-parabolic curves, we present the local parameterizations of the implicit surfaces, and we show the asymptotic behavior of the principal curvatures and directions by using the parameterization. We also present height and distance squared functions on implicit surfaces in the appendix.
{"title":"PARABOLIC, RIDGE AND SUB-PARABOLIC CURVES ON IMPLICIT SURFACES WITH SINGULARITIES","authors":"Masaru Hasegawa","doi":"10.18910/67009","DOIUrl":"https://doi.org/10.18910/67009","url":null,"abstract":"We study parabolic, ridge and sub-parabolic curves on implicit surfaces defined by smooth functions R-equivalent to A1 -singularity. To investigate ridge and sub-parabolic curves, we present the local parameterizations of the implicit surfaces, and we show the asymptotic behavior of the principal curvatures and directions by using the parameterization. We also present height and distance squared functions on implicit surfaces in the appendix.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2017-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49600401","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":"Order of the canonical vector bundle over configuration spaces of projective spaces","authors":"S. Ren","doi":"10.18910/67003","DOIUrl":"https://doi.org/10.18910/67003","url":null,"abstract":"","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2017-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48507162","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":"CORRIGENDUM TO “DEFORMATIONS OF SPECIAL LEGENDRIAN SUBMANIFOLDS WITH BOUNDARY” OSAKA J. MATH. 51 (2014), 673–693","authors":"Guangcun Lu, Xiaomin Chen","doi":"10.18910/67002","DOIUrl":"https://doi.org/10.18910/67002","url":null,"abstract":"","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2017-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49445474","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}
Suppose that f and g are two elliptic quaternionic M¨obius transformations of orders m and n respectively. If the hyperbolic distance δ ( f , g ) between fix( f ) and fix( g ) satisfies cosh δ ( f , g ) ≥ cos π m cos π n + 1 sin π m sin π n , then the group (cid:3) f , g (cid:4) is discrete non-elementary and isomorphic to the free product (cid:3) f (cid:4) ∗ (cid:3) g (cid:4) .
想那f和g是二号elliptic quaternionic M¨obius transformations of命令M和n respectively。如果《fi之间距离hyperbolicδ(f, g) x (f)条和cosh xfi萨蒂(g)fi冰δ(f, g)≥因为ππn + 1,因为sinπm辛πn,然后《集团(cid 4: 3) f, g (cid)是discrete non-elementary和同构免费广告(cid》:3)f (cid 3: 4)∗(cid) g (cid: 4)。
{"title":"FREE PRODUCT OF TWO ELLIPTIC QUATERNIONIC MÖBIUS TRANSFORMATIONS","authors":"W. Cao","doi":"10.18910/61891","DOIUrl":"https://doi.org/10.18910/61891","url":null,"abstract":"Suppose that f and g are two elliptic quaternionic M¨obius transformations of orders m and n respectively. If the hyperbolic distance δ ( f , g ) between fix( f ) and fix( g ) satisfies cosh δ ( f , g ) ≥ cos π m cos π n + 1 sin π m sin π n , then the group (cid:3) f , g (cid:4) is discrete non-elementary and isomorphic to the free product (cid:3) f (cid:4) ∗ (cid:3) g (cid:4) .","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2017-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47044036","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 consider a diffusion processes ${ X_t }$ on an interval in the natural scale. Some results are known under which ${ X_t }$ is a martingale, and we give simple and analytic proofs for them.
{"title":"A remark on conditions that a diffusion in the natural scale is a martingale","authors":"Yuuki Shimizu, F. Nakano","doi":"10.18910/68358","DOIUrl":"https://doi.org/10.18910/68358","url":null,"abstract":"We consider a diffusion processes ${ X_t }$ on an interval in the natural scale. Some results are known under which ${ X_t }$ is a martingale, and we give simple and analytic proofs for them.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2017-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48601285","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 classify two-dimensional complex tori admitting automorphisms with positive entropy in terms of the entropies they exhibit. For each possible positive value of entropy, we describe the set of two-dimensional complex tori admitting automorphisms with that entropy.
{"title":"Salem numbers and automorphisms of abelian surfaces","authors":"Paul Reschke","doi":"10.18910/61899","DOIUrl":"https://doi.org/10.18910/61899","url":null,"abstract":"We classify two-dimensional complex tori admitting automorphisms with positive entropy in terms of the entropies they exhibit. For each possible positive value of entropy, we describe the set of two-dimensional complex tori admitting automorphisms with that entropy.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67886962","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 relation of non-degenerate closed 2-forms and complex structures on compact real parallelizable nilmanifolds.
研究紧实可并行零流形上非退化闭2型与复结构的关系。
{"title":"COMPLEX STRUCTURES AND NON-DEGENERATE CLOSED 2-FORMS OF COMPACT REAL PARALLELIZABLE NILMANIFOLDS","authors":"Takumi Yamada","doi":"10.18910/61889","DOIUrl":"https://doi.org/10.18910/61889","url":null,"abstract":"In this paper, we consider a relation of non-degenerate closed 2-forms and complex structures on compact real parallelizable nilmanifolds.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67886682","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":"On Jacobi forms of real weights and indices","authors":"Hiroki Aoki","doi":"10.18910/67000","DOIUrl":"https://doi.org/10.18910/67000","url":null,"abstract":"","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67897886","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 the abstract quasilinear hyperbolic equations of Kirchhoff type with perturbation. We show the existence of the wave operators and the scattering operator for small data, and that these operators are homeomorphic with respect to a suitable metric in a neighborhood of the origin. Introduction Let H be a separable complex Hilbert space H with the inner product (·, ·)H and the norm ‖ · ‖. Let A be a non-negative injective self-adjoint operator with domain D(A), and let m be a function satisfying m ∈ C2([0,∞); [m0,∞)), with a positive constant m0. Let b(t) be a C1 function on R. We consider the initial value problem of the abstract quasilinear hyperbolic equations of Kirchhoff type with perturbation u′′(t) + b(t)u′(t) + m(‖A1/2u(t)‖2)2Au(t) = 0, (0.1) u(0) = φ0, u′(0) = ψ0. (0.2) The asymptotic behavior of the solution of the equation above depends on the integrability of b with respect to t. If b(t) = (1 + t)−p with 0 ≤ p ≤ 1, it is known that the global solution of (0.1)-(0.2) exists uniquely and behaves like solutions of a corresponding parabolic equation, for small initial data (φ0, ψ0) ∈ D(A) × D(A1/2) (see Yamazaki [14] for 0 ≤ p < 1 and Ghisi and Gobbino [7] for p = 1, and see Ghisi [6] for a mildly degenerate case m(λ) = λ with γ ≥ 1 and 0 ≤ p ≤ 1). There are no result about the global solvability for large initial data in Sobolev spaces, even for the constant dissipation term. On the other hand, if b satisfies the assumptions lim t→±∞ b(t) = 0, (0.3) 〈t〉ptb′(t) ∈ L1(R), (0.4) where p ≥ 0 is a constant and 〈t〉 := (1+ |t|2)1/2, the author [15] showed the global existence of a solution for small data in some class and showed the following (see Theorems B and C): 2010 Mathematics Subject Classification. Primary 35L72; Secondary 35L90.
{"title":"Scattering for quasilinear hyperbolic equations of Kirchhoff type with perturbation","authors":"T. Yamazaki","doi":"10.18910/61904","DOIUrl":"https://doi.org/10.18910/61904","url":null,"abstract":"This paper is concerned with the abstract quasilinear hyperbolic equations of Kirchhoff type with perturbation. We show the existence of the wave operators and the scattering operator for small data, and that these operators are homeomorphic with respect to a suitable metric in a neighborhood of the origin. Introduction Let H be a separable complex Hilbert space H with the inner product (·, ·)H and the norm ‖ · ‖. Let A be a non-negative injective self-adjoint operator with domain D(A), and let m be a function satisfying m ∈ C2([0,∞); [m0,∞)), with a positive constant m0. Let b(t) be a C1 function on R. We consider the initial value problem of the abstract quasilinear hyperbolic equations of Kirchhoff type with perturbation u′′(t) + b(t)u′(t) + m(‖A1/2u(t)‖2)2Au(t) = 0, (0.1) u(0) = φ0, u′(0) = ψ0. (0.2) The asymptotic behavior of the solution of the equation above depends on the integrability of b with respect to t. If b(t) = (1 + t)−p with 0 ≤ p ≤ 1, it is known that the global solution of (0.1)-(0.2) exists uniquely and behaves like solutions of a corresponding parabolic equation, for small initial data (φ0, ψ0) ∈ D(A) × D(A1/2) (see Yamazaki [14] for 0 ≤ p < 1 and Ghisi and Gobbino [7] for p = 1, and see Ghisi [6] for a mildly degenerate case m(λ) = λ with γ ≥ 1 and 0 ≤ p ≤ 1). There are no result about the global solvability for large initial data in Sobolev spaces, even for the constant dissipation term. On the other hand, if b satisfies the assumptions lim t→±∞ b(t) = 0, (0.3) 〈t〉ptb′(t) ∈ L1(R), (0.4) where p ≥ 0 is a constant and 〈t〉 := (1+ |t|2)1/2, the author [15] showed the global existence of a solution for small data in some class and showed the following (see Theorems B and C): 2010 Mathematics Subject Classification. Primary 35L72; Secondary 35L90.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67886854","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":"L'anneau de cohomologie des variétés de Seifert non-orientables","authors":"A. Bauval, C. Hayat","doi":"10.18910/61909","DOIUrl":"https://doi.org/10.18910/61909","url":null,"abstract":"","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67887019","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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