In the present paper, we study the anabelian geometry of mixedcharacteristic local fields by an algorithmic approach. We begin by discussing some generalities on log-shells of mixed-characteristic local fields. One main topic of this discussion is the di¤erence between the log-shell and the ring of integers. This discussion concerning log-shells allows one to establish mono-anabelian reconstruction algorithms for constructing some objects related to the p-adic valuations. Next, we consider open homomorphisms between profinite groups of MLF-type. This consideration leads us to a bi-anabelian result for absolutely unramified mixed-characteristic local fields. Next, we establish some mono-anabelian reconstruction algorithms related to each of absolutely abelian mixed-characteristic local fields, mixed-characteristic local fields of degree one, and Galois-specifiable mixed-characteristic local fields. For instance, we give a mono-anabelian reconstruction algorithm for constructing the Norm map with respect to the finite extension determined by the uniquely determined minimal mixed-characteristic local subfield. Finally, we apply various results of the present paper to prove some facts concerning outer automorphisms of the absolute Galois groups of mixedcharacteristic local fields that arise from field automorphisms of the mixed-characteristic local fields.
{"title":"Topics in the anabelian geometry of mixed-characteristic local fields","authors":"Yuichiro Hoshi","doi":"10.32917/hmj/1573787035","DOIUrl":"https://doi.org/10.32917/hmj/1573787035","url":null,"abstract":"In the present paper, we study the anabelian geometry of mixedcharacteristic local fields by an algorithmic approach. We begin by discussing some generalities on log-shells of mixed-characteristic local fields. One main topic of this discussion is the di¤erence between the log-shell and the ring of integers. This discussion concerning log-shells allows one to establish mono-anabelian reconstruction algorithms for constructing some objects related to the p-adic valuations. Next, we consider open homomorphisms between profinite groups of MLF-type. This consideration leads us to a bi-anabelian result for absolutely unramified mixed-characteristic local fields. Next, we establish some mono-anabelian reconstruction algorithms related to each of absolutely abelian mixed-characteristic local fields, mixed-characteristic local fields of degree one, and Galois-specifiable mixed-characteristic local fields. For instance, we give a mono-anabelian reconstruction algorithm for constructing the Norm map with respect to the finite extension determined by the uniquely determined minimal mixed-characteristic local subfield. Finally, we apply various results of the present paper to prove some facts concerning outer automorphisms of the absolute Galois groups of mixedcharacteristic local fields that arise from field automorphisms of the mixed-characteristic local fields.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43777673","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":"A note on the products $alpha_1beta_2gamma_t$ and $beta_1^{r+1}beta_2gamma_t$ in the stable homotopy of spheres","authors":"Hiroki Okajima, K. Shimomura","doi":"10.32917/hmj/1573787034","DOIUrl":"https://doi.org/10.32917/hmj/1573787034","url":null,"abstract":"","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47026245","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 study a prescribed mean curvature problem where we seek a surface whose mean curvature vector coincides with the normal component of a given vector field. We prove that the problem has a solution near a graphical minimal surface if the prescribed vector field is sufficiently small in a dimensionally sharp Sobolev norm.
{"title":"The Dirichlet problem for a prescribed mean curvature equation","authors":"Y. Tsukamoto","doi":"10.32917/hmj/1607396492","DOIUrl":"https://doi.org/10.32917/hmj/1607396492","url":null,"abstract":"We study a prescribed mean curvature problem where we seek a surface whose mean curvature vector coincides with the normal component of a given vector field. We prove that the problem has a solution near a graphical minimal surface if the prescribed vector field is sufficiently small in a dimensionally sharp Sobolev norm.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42750332","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 bstract . We study the global-in-time existence and the asymptotic behavior of solutions to a one-dimensional chemotaxis system presented by Deneubourg (Insectes Sociaux 24 (1977)). The system models the self-organized nest construction process of social insects. In the limit as a time-scale coe‰cient tends to 0, the Deneubourg model reduces to a parabolic-parabolic Keller-Segel system with linear degradation. We first show the global-in-time existence of solutions. We next define the dynamical system of solutions and construct the global attractor. In addition, under the assumption of a large resting rate of worker insects, we construct a Lyapunov functional for the unique homogeneous equilibrium, which indicates that the global attractor consists only of the equilibrium.
{"title":"Global attractor and Lyapunov function for one-dimensional Deneubourg chemotaxis system","authors":"Kanako Noda, Koichi Osaki","doi":"10.32917/HMJ/1564106547","DOIUrl":"https://doi.org/10.32917/HMJ/1564106547","url":null,"abstract":"A bstract . We study the global-in-time existence and the asymptotic behavior of solutions to a one-dimensional chemotaxis system presented by Deneubourg (Insectes Sociaux 24 (1977)). The system models the self-organized nest construction process of social insects. In the limit as a time-scale coe‰cient tends to 0, the Deneubourg model reduces to a parabolic-parabolic Keller-Segel system with linear degradation. We first show the global-in-time existence of solutions. We next define the dynamical system of solutions and construct the global attractor. In addition, under the assumption of a large resting rate of worker insects, we construct a Lyapunov functional for the unique homogeneous equilibrium, which indicates that the global attractor consists only of the equilibrium.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41934057","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 G be an exponential solvable Lie group and H a connected Lie subgroup of G. Given any discontinuous group K for the homogeneous space M=G/H and any deformation of K, deformation of discrete subgroups may destroy proper discontinuity of the action on M as H is not compact (except the case when it is trivial). To interpret this phenomenon in the case when G is a 3-step nilpotent, we provide a layering of Kobayashi's deformation space T (K; G; H) into Hausdorff spaces, which depends upon the dimensions of G-adjoint orbits of the corresponding parameter space. This allows us to establish a Hausdorffness theorem for T (k; G; H).
{"title":"Some Problems of deformations on three-step nilpotent Lie groups","authors":"A. Baklouti, M. Boussoffara, I. Kedim","doi":"10.32917/HMJ/1564106545","DOIUrl":"https://doi.org/10.32917/HMJ/1564106545","url":null,"abstract":"Let G be an exponential solvable Lie group and H a connected Lie subgroup of G. Given any discontinuous group K for the homogeneous space M=G/H and any deformation of K, deformation of discrete subgroups may destroy proper discontinuity of the action on M as H is not compact (except the case \u0000when it is trivial). To interpret this phenomenon in the case when G is a 3-step nilpotent, we provide a layering of Kobayashi's deformation space T (K; G; H) into Hausdorff spaces, which depends upon the dimensions of G-adjoint orbits of the corresponding parameter space. This allows us to establish a Hausdorffness theorem for T (k; G; H).","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.32917/HMJ/1564106545","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44707300","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 bstract . Masser and others have constructed sequences of ‘‘near miss’’ abc -triples, i.e., triples of relatively prime rational integers ð a ; b ; c Þ that asymptotically come close to violating the inequality that appears in the abc Conjecture. In the present paper, we show various partial generalizations of Masser’s result to arbitrary number fields.
{"title":"Near miss $abc$-triples in general number fields","authors":"Y. Kawaguchi","doi":"10.32917/HMJ/1564106548","DOIUrl":"https://doi.org/10.32917/HMJ/1564106548","url":null,"abstract":"A bstract . Masser and others have constructed sequences of ‘‘near miss’’ abc -triples, i.e., triples of relatively prime rational integers ð a ; b ; c Þ that asymptotically come close to violating the inequality that appears in the abc Conjecture. In the present paper, we show various partial generalizations of Masser’s result to arbitrary number fields.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49017090","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":"A regularity criterion for a density-dependent incompressible liquid crystals model with vacuum","authors":"Jishan Fan, B. Samet, Yong Zhou","doi":"10.32917/HMJ/1554516040","DOIUrl":"https://doi.org/10.32917/HMJ/1554516040","url":null,"abstract":"","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.32917/HMJ/1554516040","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45496197","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}
Hiroki Watanabe, Masashi Hyodo, Yuki Yamada, T. Seo
The Euclidean distance-based classifier is often used to classify an observation into one of several populations in high-dimensional data. One of the most important problems in discriminant analysis is estimating the probability of misclassification. In this paper, we propose a consistent estimator of misclassification probabilities when the dimension of the vector, p, may exceed the sample size, N , and the underlying distribution need not necessarily be normal. A new estimator of quadratic form is also obtained as a by-product. Finally, we numerically verify the high accuracy of our proposed estimator in finite sample applications, inclusive of high-dimensional scenarios. AMS 2000 subject classification: 62H30, 41A60.
{"title":"Estimation of misclassification probability for a distance-based classifier in high-dimensional data","authors":"Hiroki Watanabe, Masashi Hyodo, Yuki Yamada, T. Seo","doi":"10.32917/HMJ/1564106544","DOIUrl":"https://doi.org/10.32917/HMJ/1564106544","url":null,"abstract":"The Euclidean distance-based classifier is often used to classify an observation into one of several populations in high-dimensional data. One of the most important problems in discriminant analysis is estimating the probability of misclassification. In this paper, we propose a consistent estimator of misclassification probabilities when the dimension of the vector, p, may exceed the sample size, N , and the underlying distribution need not necessarily be normal. A new estimator of quadratic form is also obtained as a by-product. Finally, we numerically verify the high accuracy of our proposed estimator in finite sample applications, inclusive of high-dimensional scenarios. AMS 2000 subject classification: 62H30, 41A60.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":"1 1","pages":""},"PeriodicalIF":0.2,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69465257","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 characteristic $p>0$ and for $q$ a power of $p$, we compute the number of nonplanar rational curves of arbitrary degrees on a smooth Hermitian surface of degree $q+1$ under the assumption that the curves have a parametrization given by polynomials with at most $4$ terms. It is shown that a smooth Hermitian cubic surface contains infinitely many rational curves of degree $3$ and $6$. On the other hand, for all other cases the numbers of curves are finite and they are exactly determined. Further such rational curves are given explicitly up to projective isomorphism and their smoothness are checked.
{"title":"Rational curves on a smooth Hermitian surface","authors":"Norifumi Ojiro","doi":"10.32917/hmj/1554516042","DOIUrl":"https://doi.org/10.32917/hmj/1554516042","url":null,"abstract":"In characteristic $p>0$ and for $q$ a power of $p$, we compute the number of nonplanar rational curves of arbitrary degrees on a smooth Hermitian surface of degree $q+1$ under the assumption that the curves have a parametrization given by polynomials with at most $4$ terms. It is shown that a smooth Hermitian cubic surface contains infinitely many rational curves of degree $3$ and $6$. On the other hand, for all other cases the numbers of curves are finite and they are exactly determined. Further such rational curves are given explicitly up to projective isomorphism and their smoothness are checked.","PeriodicalId":55054,"journal":{"name":"Hiroshima Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.2,"publicationDate":"2018-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41892024","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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