{"title":"Structure of the associated groups of quandles","authors":"Toshiyuki Akita, Aoi Hasegawa, Masayoshi Tanno","doi":"10.2996/kmj45205","DOIUrl":"https://doi.org/10.2996/kmj45205","url":null,"abstract":"","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47000954","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}
S. Mochizuki, I. Fesenko, Yuichiro Hoshi, Arata Minamide, Wojciech Porowski
In the final paper of a series of papers concerning interuniversal Teichmüller theory, Mochizuki verified various numerically non-effective versions of the Vojta, ABC, and Szpiro Conjectures over number fields. In the present paper, we obtain various numerically effective versions of Mochizuki’s results. In order to obtain these results, we first establish a version of the theory of étale theta functions that functions properly at arbitrary bad places, i.e., even bad places that divide the prime “2”. We then proceed to discuss how such a modified version of the theory of étale theta functions affects inter-universal Teichmüller theory. Finally, by applying our slightly modified version of inter-universal Teichmüller theory, together with various explicit estimates concerning heights, the j-invariants of “arithmetic” elliptic curves, and the prime number theorem, we verify the numerically effective versions of Mochizuki’s results referred to above. These numerically effective versions imply effective diophantine results such as an effective version of the ABC inequality over mono-complex number fields [i.e., the rational number field or an imaginary quadratic field] and effective versions of conjectures of Szpiro. We also obtain an explicit estimate concerning “Fermat’s Last Theorem” (FLT) — i.e., to the effect that FLT holds for prime exponents > 1.615 · 10 — which is sufficient, in light of a numerical result of Coppersmith, to give an alternative proof of the first case of FLT. In the second case of FLT, if one combines the techniques of the present paper with a recent estimate due to Mihăilescu and Rassias, then the lower bound “1.615 · 10” can be improved to “257”. This estimate, combined with a classical result of Vandiver, yields an alternative proof of the second case of FLT. In particular, the results of the present paper, combined with the results of Vandiver, Coppersmith, and Mihăilescu-Rassias, yield an unconditional new alternative proof of Fermat’s Last Theorem.
{"title":"Explicit estimates in inter-universal Teichmüller theory","authors":"S. Mochizuki, I. Fesenko, Yuichiro Hoshi, Arata Minamide, Wojciech Porowski","doi":"10.2996/kmj45201","DOIUrl":"https://doi.org/10.2996/kmj45201","url":null,"abstract":"In the final paper of a series of papers concerning interuniversal Teichmüller theory, Mochizuki verified various numerically non-effective versions of the Vojta, ABC, and Szpiro Conjectures over number fields. In the present paper, we obtain various numerically effective versions of Mochizuki’s results. In order to obtain these results, we first establish a version of the theory of étale theta functions that functions properly at arbitrary bad places, i.e., even bad places that divide the prime “2”. We then proceed to discuss how such a modified version of the theory of étale theta functions affects inter-universal Teichmüller theory. Finally, by applying our slightly modified version of inter-universal Teichmüller theory, together with various explicit estimates concerning heights, the j-invariants of “arithmetic” elliptic curves, and the prime number theorem, we verify the numerically effective versions of Mochizuki’s results referred to above. These numerically effective versions imply effective diophantine results such as an effective version of the ABC inequality over mono-complex number fields [i.e., the rational number field or an imaginary quadratic field] and effective versions of conjectures of Szpiro. We also obtain an explicit estimate concerning “Fermat’s Last Theorem” (FLT) — i.e., to the effect that FLT holds for prime exponents > 1.615 · 10 — which is sufficient, in light of a numerical result of Coppersmith, to give an alternative proof of the first case of FLT. In the second case of FLT, if one combines the techniques of the present paper with a recent estimate due to Mihăilescu and Rassias, then the lower bound “1.615 · 10” can be improved to “257”. This estimate, combined with a classical result of Vandiver, yields an alternative proof of the second case of FLT. In particular, the results of the present paper, combined with the results of Vandiver, Coppersmith, and Mihăilescu-Rassias, yield an unconditional new alternative proof of Fermat’s Last Theorem.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44496933","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 $M^n$ be an $n$-dimensional complete and locally conformally flat hypersurface in the unit sphere $mathbb{S}^{n+1}$ with constant scalar curvature $n(n-1)$. We show that if the total curvature $left( int _ { M } | H | ^ { n } d v right) ^ { frac { 1 } { n } }$ of $M$ is sufficiently small, then $M^n$ is totally geodesic.
{"title":"On complete hypersurfaces with constant scalar curvature $n(n-1)$ in the unit sphere","authors":"Jinchuan Bai, Yong Luo","doi":"10.2996/kmj46104","DOIUrl":"https://doi.org/10.2996/kmj46104","url":null,"abstract":"Let $M^n$ be an $n$-dimensional complete and locally conformally flat hypersurface in the unit sphere $mathbb{S}^{n+1}$ with constant scalar curvature $n(n-1)$. We show that if the total curvature $left( int _ { M } | H | ^ { n } d v right) ^ { frac { 1 } { n } }$ of $M$ is sufficiently small, then $M^n$ is totally geodesic.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42709300","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 our previous work, we have generalized the notion of dually flat or Hessian manifold to quasi-Hessian manifold ; it admits the Hessian metric to be degenerate but possesses a particular symmetric cubic tensor (generalized Amari-Centsov tensor). Indeed, it naturally appears as a singular model in information geometry and related fields. A quasi-Hessian manifold is locally accompanied with a possibly multi-valued potential and its dual, whose graphs are called the e -wavefront and the m -wavefront respectively, together with coherent tangent bundles endowed with flat connections. In the present paper, using those connections and the metric, we give coordinate-free criteria for detecting local diffeomorphic types of e/m -wavefronts, and then derive the local normal forms of those (dual) potential functions for the e/m -wavefronts in affine flat coordinates by means of Malgrange’s division theorem. This is motivated by an early work of Ekeland on non-convex optimization and Saji-Umehara-Yamada’s work on Riemannian geometry of wavefronts. Finally, we reveal a relation of our geometric criteria with information geometric quantities of statistical manifolds.
{"title":"Local normal forms of em-wavefronts in affine flat coordinates","authors":"Naomichi Nakajima","doi":"10.2996/kmj45305","DOIUrl":"https://doi.org/10.2996/kmj45305","url":null,"abstract":". In our previous work, we have generalized the notion of dually flat or Hessian manifold to quasi-Hessian manifold ; it admits the Hessian metric to be degenerate but possesses a particular symmetric cubic tensor (generalized Amari-Centsov tensor). Indeed, it naturally appears as a singular model in information geometry and related fields. A quasi-Hessian manifold is locally accompanied with a possibly multi-valued potential and its dual, whose graphs are called the e -wavefront and the m -wavefront respectively, together with coherent tangent bundles endowed with flat connections. In the present paper, using those connections and the metric, we give coordinate-free criteria for detecting local diffeomorphic types of e/m -wavefronts, and then derive the local normal forms of those (dual) potential functions for the e/m -wavefronts in affine flat coordinates by means of Malgrange’s division theorem. This is motivated by an early work of Ekeland on non-convex optimization and Saji-Umehara-Yamada’s work on Riemannian geometry of wavefronts. Finally, we reveal a relation of our geometric criteria with information geometric quantities of statistical manifolds.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41660268","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":"Estimates for the higher eigenvalues of the drifting Laplacian on Hadamard manifolds","authors":"Xin Xiong, Lingzhong Zeng, Huihui Zhu","doi":"10.2996/kmj/kmj45109","DOIUrl":"https://doi.org/10.2996/kmj/kmj45109","url":null,"abstract":"","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46588617","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 the dimension of the global sections of adjoint bundles for quasi-polarized manifold whose anti-canonical bundle is effective, nef and big","authors":"Y. Fukuma","doi":"10.2996/kmj/kmj45101","DOIUrl":"https://doi.org/10.2996/kmj/kmj45101","url":null,"abstract":"","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44649745","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 adjoint torsion polynomial of genus one two-bridge knots","authors":"Takayuki Morifuji","doi":"10.2996/kmj/kmj45107","DOIUrl":"https://doi.org/10.2996/kmj/kmj45107","url":null,"abstract":"","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42719720","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}
Morimichi Kawasaki, M. Kimura, Shuhei Maruyama, Takahiro Matsushita, M. Mimura
In the present paper, for a pair $(G,N)$ of a group $G$ and its normal subgroup $N$, we consider the mixed commutator length $mathrm{cl}_{G,N}$ on the mixed commutator subgroup $[G,N]$. We focus on the setting of wreath products: $ (G,N)=(mathbb{Z}wr Gamma, bigoplus_{Gamma}mathbb{Z})$. Then we determine mixed commutator lengths in terms of the general rank in the sense of Malcev. As a byproduct, when an abelian group $Gamma$ is not locally cyclic, the ordinary commutator length $mathrm{cl}_G$ does not coincide with $mathrm{cl}_{G,N}$ on $[G,N]$ for the above pair. On the other hand, we prove that if $Gamma$ is locally cyclic, then for every pair $(G,N)$ such that $1to Nto Gto Gamma to 1$ is exact, $mathrm{cl}_{G}$ and $mathrm{cl}_{G,N}$ coincide on $[G,N]$. We also study the case of permutational wreath products when the group $Gamma$ belongs to a certain class related to surface groups.
{"title":"Mixed commutator lengths, wreath products and general ranks","authors":"Morimichi Kawasaki, M. Kimura, Shuhei Maruyama, Takahiro Matsushita, M. Mimura","doi":"10.2996/kmj46202","DOIUrl":"https://doi.org/10.2996/kmj46202","url":null,"abstract":"In the present paper, for a pair $(G,N)$ of a group $G$ and its normal subgroup $N$, we consider the mixed commutator length $mathrm{cl}_{G,N}$ on the mixed commutator subgroup $[G,N]$. We focus on the setting of wreath products: $ (G,N)=(mathbb{Z}wr Gamma, bigoplus_{Gamma}mathbb{Z})$. Then we determine mixed commutator lengths in terms of the general rank in the sense of Malcev. As a byproduct, when an abelian group $Gamma$ is not locally cyclic, the ordinary commutator length $mathrm{cl}_G$ does not coincide with $mathrm{cl}_{G,N}$ on $[G,N]$ for the above pair. On the other hand, we prove that if $Gamma$ is locally cyclic, then for every pair $(G,N)$ such that $1to Nto Gto Gamma to 1$ is exact, $mathrm{cl}_{G}$ and $mathrm{cl}_{G,N}$ coincide on $[G,N]$. We also study the case of permutational wreath products when the group $Gamma$ belongs to a certain class related to surface groups.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44117954","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 a Howe curve C in positive characteristic p ≥ 3 which is of genus 3 and is hyperelliptic. We will show that if C is superspecial, then its standard form is maximal or minimal over Fp2 without taking its Fp2 -form.
{"title":"On the maximality of hyperelliptic Howe curves of genus 3","authors":"R. Ohashi","doi":"10.2996/kmj45206","DOIUrl":"https://doi.org/10.2996/kmj45206","url":null,"abstract":"In this paper, we study a Howe curve C in positive characteristic p ≥ 3 which is of genus 3 and is hyperelliptic. We will show that if C is superspecial, then its standard form is maximal or minimal over Fp2 without taking its Fp2 -form.","PeriodicalId":54747,"journal":{"name":"Kodai Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47469457","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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