We explore the relationship in metric spaces between different properties related to the Besicovitch covering theorem, and also consider weak versions of doubling, in connection to the non-uniqueness of centers and radii in arbitrary metric spaces.
{"title":"Besicovitch and doubling type properties in metric spaces","authors":"J. M. Aldaz","doi":"10.14492/hokmj/2021-528","DOIUrl":"https://doi.org/10.14492/hokmj/2021-528","url":null,"abstract":"We explore the relationship in metric spaces between different properties related to the Besicovitch covering theorem, and also consider weak versions of doubling, in connection to the non-uniqueness of centers and radii in arbitrary metric spaces.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42855998","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}
Pub Date : 2019-02-01DOI: 10.14492/HOKMJ/1550480647
Takashi Hirotsu
A Châtelet surface over a field is a typical geometrically rational surface. Its Chow group of zero-cycles has been studied as an important birational invariant by many researchers since the 1970s. Recently, S. Saito and K. Sato obtained a duality between the Chow and Brauer groups from the Brauer–Manin pairing. For a Châtelet surface over a local field, we combine their result with the known calculation of the Chow group to determine the structure and generators of the Brauer group of a regular proper flat model of the surface over the integer ring of the base field.
{"title":"Brauer groups of Châtelet surfaces over local fields","authors":"Takashi Hirotsu","doi":"10.14492/HOKMJ/1550480647","DOIUrl":"https://doi.org/10.14492/HOKMJ/1550480647","url":null,"abstract":"A Châtelet surface over a field is a typical geometrically rational surface. Its Chow group of zero-cycles has been studied as an important birational invariant by many researchers since the 1970s. Recently, S. Saito and K. Sato obtained a duality between the Chow and Brauer groups from the Brauer–Manin pairing. For a Châtelet surface over a local field, we combine their result with the known calculation of the Chow group to determine the structure and generators of the Brauer group of a regular proper flat model of the surface over the integer ring of the base field.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.14492/HOKMJ/1550480647","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48780245","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}
Pub Date : 2019-02-01DOI: 10.14492/hokmj/1550480649
S. Rezaei
Let a denote an ideal in a commutative Noetherian local ring (R,m) and M a non-zero finitely generated R-module of dimension d. Let d := dim(M/aM). In this paper we calculate the annihilator of the top formal local cohomology module Fda(M). In fact, we prove that AnnR(F d a(M)) = AnnR(M/UR(a,M)), where UR(a,M) := ∪{N : N ⩽ M and dim(N/aN) < dim(M/aM)}. We give a description of UR(a,M) and we will show that AnnR(F d a(M)) = AnnR(M/ ∩pj∈AsshRM∩V(a) Nj), where 0 = ∩n j=1 Nj denotes a reduced primary decomposition of the zero submodule 0 in M and Nj is a pj-primary submodule of M , for all j = 1, . . . , n. Also, we determine the radical of the annihilator of Fda(M). We will prove that √ AnnR(Fa(M)) = AnnR(M/GR(a,M)), where GR(a,M) denotes the largest submodule of M such that AsshR(M) ∩ V(a) ⊆ AssR(M/GR(a,M)) and AsshR(M) denotes the set {p ∈ AssM : dimR/p = dimM}.
设a表示交换诺瑟局部环(R,m)中的理想，m表示维数为d的非零有限生成R模。设d:= dim(m /aM)。本文计算了上形式局部上同模Fda(M)的湮灭子。事实上，我们证明了AnnR(F d a(M)) = AnnR(M/UR(a,M))，其中UR(a,M):=∪{N: N≤M and dim(N/aN) < dim(M/aM)}。我们给出了UR(a,M)的一个描述，并且我们将证明AnnR(F d a(M)) = AnnR(M/∩pj∈AsshRM∩V(a) Nj)，其中0 =∩n j=1 Nj表示M中的零子模块0的一个约简初分解，并且Nj是M的一个pj-主子模块，对于所有j=1，…， n。同时，我们确定了Fda(M)湮灭子的原子量。我们将证明√AnnR(Fa(M)) = AnnR(M/GR(a,M))，其中GR(a,M)表示M的最大子模块，使得AssR(M)∩V(a)≤AssR(M/GR(a,M))，且AssR(M)表示集合{p∈AssM: dimR/p = dimM}。
{"title":"On the annihilators of formal local cohomology modules","authors":"S. Rezaei","doi":"10.14492/hokmj/1550480649","DOIUrl":"https://doi.org/10.14492/hokmj/1550480649","url":null,"abstract":"Let a denote an ideal in a commutative Noetherian local ring (R,m) and M a non-zero finitely generated R-module of dimension d. Let d := dim(M/aM). In this paper we calculate the annihilator of the top formal local cohomology module Fda(M). In fact, we prove that AnnR(F d a(M)) = AnnR(M/UR(a,M)), where UR(a,M) := ∪{N : N ⩽ M and dim(N/aN) < dim(M/aM)}. We give a description of UR(a,M) and we will show that AnnR(F d a(M)) = AnnR(M/ ∩pj∈AsshRM∩V(a) Nj), where 0 = ∩n j=1 Nj denotes a reduced primary decomposition of the zero submodule 0 in M and Nj is a pj-primary submodule of M , for all j = 1, . . . , n. Also, we determine the radical of the annihilator of Fda(M). We will prove that √ AnnR(Fa(M)) = AnnR(M/GR(a,M)), where GR(a,M) denotes the largest submodule of M such that AsshR(M) ∩ V(a) ⊆ AssR(M/GR(a,M)) and AsshR(M) denotes the set {p ∈ AssM : dimR/p = dimM}.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.14492/hokmj/1550480649","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42853403","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}
Pub Date : 2019-02-01DOI: 10.14492/hokmj/1550480646
E. Nakai, Tsuyoshi Yoneda
We give new viewpoints of Campanato spaces with variable growth condition for applications to the Navier-Stokes equation. Namely, we formulate a blowup criteria along maximum points of the 3D-Navier-Stokes flow in terms of stationary Euler flows and show that the properties of Campanato spaces with variable growth condition are very useful for this formulation, since variable growth condition can control the continuity and integrability of functions on the neighborhood at each point. Our criterion is different from the Beale-Kato-Majda type and Constantin-Fefferman type criterion. If geometric behavior of the velocity vector field near the maximum point has a kind of stationary Euler flow configuration up to a possible blowup time, then the solution can be extended to be the strong solution beyond the possible blowup time. As another application we also mention the Cauchy problem for the NavierStokes equation.
{"title":"Applications of Campanato spaces with variable growth condition to the Navier-Stokes equation","authors":"E. Nakai, Tsuyoshi Yoneda","doi":"10.14492/hokmj/1550480646","DOIUrl":"https://doi.org/10.14492/hokmj/1550480646","url":null,"abstract":"We give new viewpoints of Campanato spaces with variable growth condition for applications to the Navier-Stokes equation. Namely, we formulate a blowup criteria along maximum points of the 3D-Navier-Stokes flow in terms of stationary Euler flows and show that the properties of Campanato spaces with variable growth condition are very useful for this formulation, since variable growth condition can control the continuity and integrability of functions on the neighborhood at each point. Our criterion is different from the Beale-Kato-Majda type and Constantin-Fefferman type criterion. If geometric behavior of the velocity vector field near the maximum point has a kind of stationary Euler flow configuration up to a possible blowup time, then the solution can be extended to be the strong solution beyond the possible blowup time. As another application we also mention the Cauchy problem for the NavierStokes equation.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.14492/hokmj/1550480646","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44221430","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 analyze the asymptotic behavior of solutions to wave equations with strong damping terms. If the initial data belong to suitable weighted $L^1$ spaces, lower bounds for the difference between the solutions and the leading terms in the Fourier space are obtained, which implies the optimality of expanding methods and some estimates proposed in this paper.
{"title":"Optimal leading term of solutions to wave equations with strong damping terms","authors":"Hironori Michihisa","doi":"10.14492/hokmj/2018-920","DOIUrl":"https://doi.org/10.14492/hokmj/2018-920","url":null,"abstract":"We analyze the asymptotic behavior of solutions to wave equations with strong damping terms. If the initial data belong to suitable weighted $L^1$ spaces, lower bounds for the difference between the solutions and the leading terms in the Fourier space are obtained, which implies the optimality of expanding methods and some estimates proposed in this paper.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2018-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46347856","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 note, we calculate the boundary of movable cones and nef cones of the generalized Kummer 4-fold $mathrm{Km}^2(A)$ attached to an abelian surface $A$ with $mathrm{rkNS}(A) = 1$.
{"title":"Nef Cone of a Generalized Kummer 4-fold","authors":"Akira Mori","doi":"10.14492/hokmj/2018-919","DOIUrl":"https://doi.org/10.14492/hokmj/2018-919","url":null,"abstract":"In this note, we calculate the boundary of movable cones and nef cones of the generalized Kummer 4-fold $mathrm{Km}^2(A)$ attached to an abelian surface $A$ with $mathrm{rkNS}(A) = 1$.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2018-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47177506","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}
Pub Date : 2018-10-01DOI: 10.14492/HOKMJ/1537948827
R. Kakizawa
This paper deals with the global existence of weak solutions to the initial value problem for the Navier-Stokes equations in R (n ∈ Z, n ≥ 2). Concerning initial data of the form Ax + v(0), where A ∈ Mn(R) and v(0) ∈ Lσ(R), the weak solutions are properly-defined with the aid of the alternativity of the trilinear from (Ax ·∇)v ·φ. Furthermore, we construct the Leray-Hopf weak solution which satisfies not only the Navier-Stokes equations but also the energy inequality via the Galerkin approximation. From the viewpoint of quadratic forms, the Gronwall-Bellman inequality admits the uniform boundedness of the approximate solution.
{"title":"The existence of Leray-Hopf weak solutions with linear strain","authors":"R. Kakizawa","doi":"10.14492/HOKMJ/1537948827","DOIUrl":"https://doi.org/10.14492/HOKMJ/1537948827","url":null,"abstract":"This paper deals with the global existence of weak solutions to the initial value problem for the Navier-Stokes equations in R (n ∈ Z, n ≥ 2). Concerning initial data of the form Ax + v(0), where A ∈ Mn(R) and v(0) ∈ Lσ(R), the weak solutions are properly-defined with the aid of the alternativity of the trilinear from (Ax ·∇)v ·φ. Furthermore, we construct the Leray-Hopf weak solution which satisfies not only the Navier-Stokes equations but also the energy inequality via the Galerkin approximation. From the viewpoint of quadratic forms, the Gronwall-Bellman inequality admits the uniform boundedness of the approximate solution.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45521308","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}
Pub Date : 2018-10-01DOI: 10.14492/HOKMJ/1537948835
S. Ohno, T. Sakai, H. Urakawa
On a smooth foliated map from a complete, possibly non-compact, foliated Riemannian manifold into another foliated Riemannian manifold of which transversal sectional curvature is non-positive, we will show that, if it is transversally biharmonic and has the finite energy and finite bienergy, then it is transversally harmonic.
{"title":"Rigidity of transversally biharmonic maps between foliated Riemannian manifolds","authors":"S. Ohno, T. Sakai, H. Urakawa","doi":"10.14492/HOKMJ/1537948835","DOIUrl":"https://doi.org/10.14492/HOKMJ/1537948835","url":null,"abstract":"On a smooth foliated map from a complete, possibly non-compact, foliated Riemannian manifold into another foliated Riemannian manifold of which transversal sectional curvature is non-positive, we will show that, if it is transversally biharmonic and has the finite energy and finite bienergy, then it is transversally harmonic.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.14492/HOKMJ/1537948835","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43242116","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}
Pub Date : 2018-10-01DOI: 10.14492/hokmj/1537948828
R. Farwig, R. Schulz, Y. Taniuchi
The nonstationary Navier-Stokes system for a viscous, incompressible fluid influenced by a Coriolis force in the whole space R3 is considered at large distances. The solvability of the corresponding integral equations of these equations in weighted L∞-spaces is established. Furthermore, the leading terms of the asymptotic profile of the solution at fixed time t > 0 for |x| > t and far from the axis of rotation are investigated.
{"title":"Spatial Asymptotic Profiles of Solutions to the Navier-Stokes System in a Rotating Frame with Fast Decaying Data","authors":"R. Farwig, R. Schulz, Y. Taniuchi","doi":"10.14492/hokmj/1537948828","DOIUrl":"https://doi.org/10.14492/hokmj/1537948828","url":null,"abstract":"The nonstationary Navier-Stokes system for a viscous, incompressible fluid influenced by a Coriolis force in the whole space R3 is considered at large distances. The solvability of the corresponding integral equations of these equations in weighted L∞-spaces is established. Furthermore, the leading terms of the asymptotic profile of the solution at fixed time t > 0 for |x| > t and far from the axis of rotation are investigated.","PeriodicalId":55051,"journal":{"name":"Hokkaido Mathematical Journal","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2018-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.14492/hokmj/1537948828","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48489419","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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