We consider iterated function systems on the unit interval generated by two contractive similarity transformations with the same similarity ratio. When the ratio is greater than or equal to $1/2$, the limit set is the interval itself and the code map is not one-to-one. We study the set of points of the limit set having unique addresses. We obtain a formula for the Hausdorff dimension of the set when the similarity ratio belongs to certain intervals by applying the concept of graph directed Markov system.
{"title":"THE HAUSDORFF DIMENSION OF THE REGION OF MULTIPLICITY ONE OF OVERLAPPING ITERATED FUNCTION SYSTEMS ON THE INTERVAL","authors":"Kengo Shimomura","doi":"10.18910/79428","DOIUrl":"https://doi.org/10.18910/79428","url":null,"abstract":"We consider iterated function systems on the unit interval generated by two contractive similarity transformations with the same similarity ratio. When the ratio is greater than or equal to $1/2$, the limit set is the interval itself and the code map is not one-to-one. We study the set of points of the limit set having unique addresses. We obtain a formula for the Hausdorff dimension of the set when the similarity ratio belongs to certain intervals by applying the concept of graph directed Markov system.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44958454","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}
First we introduce the notion of Killing structure Jacobi operator for real hypersurfaces in the complex hyperbolic quadric ${Q^m}^*=SO^0_{2,m}/SO_2 SO_m$ . Next we give a complete classification of real hypersurfaces in ${Q^m}^*=SO^0_{2,m}/SO_2 SO_m$ with Killing structure Jacobi operator.
{"title":"Real hypersurfaces with Killing structure Jacobi operator in the complex hyperbolic quadric","authors":"Jin Suh Young","doi":"10.18910/78987","DOIUrl":"https://doi.org/10.18910/78987","url":null,"abstract":"First we introduce the notion of Killing structure Jacobi operator for real hypersurfaces in the complex hyperbolic quadric ${Q^m}^*=SO^0_{2,m}/SO_2 SO_m$ . Next we give a complete classification of real hypersurfaces in ${Q^m}^*=SO^0_{2,m}/SO_2 SO_m$ with Killing structure Jacobi operator.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67930260","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 twisted Alexander polynomial is defined as a rational function, not necessarily a polynomial. It is shown that for a ribbon 2-knot, the twisted Alexander polynomial associated to an irreducible representation of the knot group to SL(2 , F ) is always a polynomial. Further-more, the twisted Alexander polynomial of a fibered ribbon 2-knot of 1-fusion has the lowest and highest degree coe ffi cients 1 with breadth 2 m − 2, where m is the breadth of its Alexander polynomial.
{"title":"Twisted alexander polynomial of a ribbon 2-knot of 1-fusion","authors":"T. Kanenobu, Toshio Sumi","doi":"10.18910/77230","DOIUrl":"https://doi.org/10.18910/77230","url":null,"abstract":"The twisted Alexander polynomial is defined as a rational function, not necessarily a polynomial. It is shown that for a ribbon 2-knot, the twisted Alexander polynomial associated to an irreducible representation of the knot group to SL(2 , F ) is always a polynomial. Further-more, the twisted Alexander polynomial of a fibered ribbon 2-knot of 1-fusion has the lowest and highest degree coe ffi cients 1 with breadth 2 m − 2, where m is the breadth of its Alexander polynomial.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43811467","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 investigate the local differential geometric invariants of cuspidal edge and swallowtail from the view point of singularity theory. We introduce finite type invariants of such singularities (see Remark 1.5 and Theorem 2.11) based on certain normal forms for cuspidal edge and swallowtail. Then we discuss several geometric aspects based on our normal form. We also present several asymptotic formulas concerning our invariants with respect to Gauss curvature and mean curvature. Typical examples of wave fronts are parallel surfaces of a regular surface in the 3dimensional Euclidean space, and it is well-known that such surfaces may have several singularities like cuspidal edge and swallowtail. Singularity types of parallel surfaces are investigated in [3], and the next interest is to investigate local differential geometries of such singularities. There are several attempts to describe them. For instance, K. Saji, M. Umehara, and K. Yamada ([12]) defined the notion of singular curvature κs and normal curvature κν of cuspidal edge, and, later, K. Saji and L. Martins ([7]) described all invariants up to order 3. It is clear that there are more differential geometric invariants in higher order terms, and to describe all such invariants up to finite order is one motivation of the paper. Since Gauss curvature and mean curvature are often diverge at singularities and we are interested in their asymptotic behaviors near a singularity in terms of our invariants. We are going to describe their asymptotic behaviors of our local differential geometric invariants of cuspidal edge near swallowtail. An ideas of singularity theory is to reduce a given map-germ (R2, 0) → (R3, 0) to certain normal form (see [9], for example). Their normal forms are obtained up to -equivalence where is the group of coordinate changes of the source and the target. In that context, we reduce a given map-germ to one of normal forms in the list there, composing certain coordinate changes of the source and the target. For differential geometric purpose, general coordinate changes of the target are too rough, since they do not preserve differential geometric properties, and we should restrict the coordinate change of the target to the motion group. From this point, we will consider the product group of coordinate change of the source with the motion group of the target (the rotation group when we consider map-germs) and we introduce a normal form for cuspidal edge (see (1.1)) and swallowtail (Theorem 2.4) by the equivalence relation defined by this group. We believe that this is a powerful method to investigate singular surfaces, since this unable us to describe all differential geometric 2010 Mathematics Subject Classification. Primary 57R45; Secondary 53A05. Dedicated to Professor Takashi Nishimura on the occasion of his 60th birthday.
{"title":"Local differential geometry of cuspidal edge and swallowtail","authors":"Toshizumi Fukui","doi":"10.18910/77239","DOIUrl":"https://doi.org/10.18910/77239","url":null,"abstract":"We investigate the local differential geometric invariants of cuspidal edge and swallowtail from the view point of singularity theory. We introduce finite type invariants of such singularities (see Remark 1.5 and Theorem 2.11) based on certain normal forms for cuspidal edge and swallowtail. Then we discuss several geometric aspects based on our normal form. We also present several asymptotic formulas concerning our invariants with respect to Gauss curvature and mean curvature. Typical examples of wave fronts are parallel surfaces of a regular surface in the 3dimensional Euclidean space, and it is well-known that such surfaces may have several singularities like cuspidal edge and swallowtail. Singularity types of parallel surfaces are investigated in [3], and the next interest is to investigate local differential geometries of such singularities. There are several attempts to describe them. For instance, K. Saji, M. Umehara, and K. Yamada ([12]) defined the notion of singular curvature κs and normal curvature κν of cuspidal edge, and, later, K. Saji and L. Martins ([7]) described all invariants up to order 3. It is clear that there are more differential geometric invariants in higher order terms, and to describe all such invariants up to finite order is one motivation of the paper. Since Gauss curvature and mean curvature are often diverge at singularities and we are interested in their asymptotic behaviors near a singularity in terms of our invariants. We are going to describe their asymptotic behaviors of our local differential geometric invariants of cuspidal edge near swallowtail. An ideas of singularity theory is to reduce a given map-germ (R2, 0) → (R3, 0) to certain normal form (see [9], for example). Their normal forms are obtained up to -equivalence where is the group of coordinate changes of the source and the target. In that context, we reduce a given map-germ to one of normal forms in the list there, composing certain coordinate changes of the source and the target. For differential geometric purpose, general coordinate changes of the target are too rough, since they do not preserve differential geometric properties, and we should restrict the coordinate change of the target to the motion group. From this point, we will consider the product group of coordinate change of the source with the motion group of the target (the rotation group when we consider map-germs) and we introduce a normal form for cuspidal edge (see (1.1)) and swallowtail (Theorem 2.4) by the equivalence relation defined by this group. We believe that this is a powerful method to investigate singular surfaces, since this unable us to describe all differential geometric 2010 Mathematics Subject Classification. Primary 57R45; Secondary 53A05. Dedicated to Professor Takashi Nishimura on the occasion of his 60th birthday.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43775627","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":"Equigeodesics on generalized flag manifolds with ${rm G}_2$-type $mathfrak{t}$-roots","authors":"Marina Statha","doi":"10.18910/77235","DOIUrl":"https://doi.org/10.18910/77235","url":null,"abstract":"","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45600261","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 prime number p ≡ 3 mod 4, we write p = 2 n (cid:2) f + 1 for some power (cid:2) f of an odd prime number (cid:2) and an odd integer n with (cid:2) (cid:2) n . For 0 ≤ t ≤ f , let K t be the imaginary subfield of Q ( ζ p ) of degree 2 (cid:2) t and let h − t be the relative class number of K t . We show that for n = 1 (resp. n ≥ 3), a prime number r does not divide the ratio h − t / h − t − 1 when r is a primitive root modulo (cid:2) 2 and r ≥ (cid:2) f − t − 1 (resp. r ≥ ( n − 2) (cid:2) f − t + 1). In particular, for n = 1 or 3, the ratio h − f / h − f − 1 at the top is not divisible by r whenever r is a primitive root modulo (cid:2) 2 . Further, we show that the (cid:2) -part of h − t / h − t − 1 stabilizes for “large” t under some assumption.
对于素数p lect 3 mod 4，我们为奇数素数（cid:2）和奇数整数n的某个幂（cid:2）f写p=2n（cid:2）f+1。对于0≤t≤f，设KT为2（cid:2）t阶Q（ζp）的虚子域，设h−t为KT的相对类数。我们证明，对于n=1（分别为n≥3），当r是基根模（cid:2）2且r≥（cid:2）f−t−1时，素数r不除以比率h−t/h−t−1。特别是，对于n=1或3，当r是模（cid:2）2的原始根时，顶部的比率h−f/h−f−1不可被r整除。此外，我们证明了在某种假设下，h−t/h−t−1的（cid:2）-部分对“大”t稳定。
{"title":"Relative class numbers inside the $p$th cyclotomic field","authors":"H. Ichimura","doi":"10.18910/77238","DOIUrl":"https://doi.org/10.18910/77238","url":null,"abstract":"For a prime number p ≡ 3 mod 4, we write p = 2 n (cid:2) f + 1 for some power (cid:2) f of an odd prime number (cid:2) and an odd integer n with (cid:2) (cid:2) n . For 0 ≤ t ≤ f , let K t be the imaginary subfield of Q ( ζ p ) of degree 2 (cid:2) t and let h − t be the relative class number of K t . We show that for n = 1 (resp. n ≥ 3), a prime number r does not divide the ratio h − t / h − t − 1 when r is a primitive root modulo (cid:2) 2 and r ≥ (cid:2) f − t − 1 (resp. r ≥ ( n − 2) (cid:2) f − t + 1). In particular, for n = 1 or 3, the ratio h − f / h − f − 1 at the top is not divisible by r whenever r is a primitive root modulo (cid:2) 2 . Further, we show that the (cid:2) -part of h − t / h − t − 1 stabilizes for “large” t under some assumption.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44930480","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}
Our aim in this paper is to deal with boundary limits of monotone Sobolev functions for the double phase functional Φp,q(x, t) = t p + (b(x)t)q in the unit ball B of Rn, where 1 < p < q < ∞ and b(·) is a non-negative bounded function on B which is Hölder continuous of order θ ∈ (0, 1].
{"title":"BOUNDARY LIMITS OF MONOTONE SOBOLEV FUNCTIONS FOR DOUBLE PHASE FUNCTIONALS","authors":"Y. Mizuta, T. Shimomura","doi":"10.18910/77232","DOIUrl":"https://doi.org/10.18910/77232","url":null,"abstract":"Our aim in this paper is to deal with boundary limits of monotone Sobolev functions for the double phase functional Φp,q(x, t) = t p + (b(x)t)q in the unit ball B of Rn, where 1 < p < q < ∞ and b(·) is a non-negative bounded function on B which is Hölder continuous of order θ ∈ (0, 1].","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48100738","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}
An extrinsic symmetric space is a submanifold M ⊂ V = Rn which is kept invariant by the reflection sx along every normal space NxM. An extrinsic symmetric subspace is a connected component M′ of the intersection M ∩ V ′ for some subspace V ′ ⊂ V which is sx-invariant for any x ∈ M′. We give an algebraic charactrization of all such subspaces V ′.
{"title":"Extrinsic symmetric subspaces","authors":"J. Eschenburg, M. Tanaka","doi":"10.18910/76678","DOIUrl":"https://doi.org/10.18910/76678","url":null,"abstract":"An extrinsic symmetric space is a submanifold M ⊂ V = Rn which is kept invariant by the reflection sx along every normal space NxM. An extrinsic symmetric subspace is a connected component M′ of the intersection M ∩ V ′ for some subspace V ′ ⊂ V which is sx-invariant for any x ∈ M′. We give an algebraic charactrization of all such subspaces V ′.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49140775","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 prove three conjectures of Tsukano about exponential sums stated in his Master’s thesis written at Osaka University. These conjectures are variations of earlier conjectures made by Lee and Weintraub which were first proved by Ibukiyama and Saito.
{"title":"THE TSUKANO CONJECTURES ON EXPONENTIAL SUMS","authors":"Brad Isaacson","doi":"10.18910/76672","DOIUrl":"https://doi.org/10.18910/76672","url":null,"abstract":"We prove three conjectures of Tsukano about exponential sums stated in his Master’s thesis written at Osaka University. These conjectures are variations of earlier conjectures made by Lee and Weintraub which were first proved by Ibukiyama and Saito.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47308705","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}
A handlebody-knot is a handlebody embedded in the 3-sphere. We introduce an invariant for handlebody-knots derived from their Alexander polynomials. The value of the invariant is a vertex-weighted graph. As an application, we describe a sufficient condition for a handlebody-knot to be irreducible and a necessary condition for a link to be a constituent link of a handlebody-knot.
{"title":"AN INVARIANT DERIVED FROM THE ALEXANDER POLYNOMIAL FOR HANDLEBODY-KNOTS","authors":"S. Okazaki","doi":"10.18910/76683","DOIUrl":"https://doi.org/10.18910/76683","url":null,"abstract":"A handlebody-knot is a handlebody embedded in the 3-sphere. We introduce an invariant for handlebody-knots derived from their Alexander polynomials. The value of the invariant is a vertex-weighted graph. As an application, we describe a sufficient condition for a handlebody-knot to be irreducible and a necessary condition for a link to be a constituent link of a handlebody-knot.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48127416","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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