Welded knotted objects are a combinatorial extension of knot theory, which can be used as a tool for studying ribbon surfaces in 4-space. A finite type invariant theory for ribbon knotted surfaces was developped by Kanenobu, Habiro and Shima, and this paper proposes a study of these invariants, using welded objects. Specifically, we study welded string links up to wk-equivalence, which is an equivalence relation introduced by Yasuhara and the second author in connection with finite type theory. In low degrees, we show that this relation characterizes the information contained by finite type invariants. We also study the algebraic structure of welded string links up to wk-equivalence. All results have direct corollaries for ribbon knotted surfaces.
{"title":"On Finite Type Invariants of Welded String Links and Ribbon Tubes","authors":"Adrien Casejuane, Jean-Baptiste Meilhan","doi":"10.3836/tjm/1502179380","DOIUrl":"https://doi.org/10.3836/tjm/1502179380","url":null,"abstract":"Welded knotted objects are a combinatorial extension of knot theory, which can be used as a tool for studying ribbon surfaces in 4-space. A finite type invariant theory for ribbon knotted surfaces was developped by Kanenobu, Habiro and Shima, and this paper proposes a study of these invariants, using welded objects. Specifically, we study welded string links up to wk-equivalence, which is an equivalence relation introduced by Yasuhara and the second author in connection with finite type theory. In low degrees, we show that this relation characterizes the information contained by finite type invariants. We also study the algebraic structure of welded string links up to wk-equivalence. All results have direct corollaries for ribbon knotted surfaces.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45537971","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 study is based on the paper [1]. We give the formula to extract the position and the shape of the defect D generated in the object (conductor) Ω from the observation data on the boundary ∂Ω for the magnetic Schrödinger operator by using the enclosure method proposed by Ikehata [2]. We show a reconstruction formula of the convex hull of the defect D from the observed data, assuming certain higher regularity for the potentials of the magnetic Schrödinger operator, under the Dirichlet condition or the Robin condition on the boundary ∂D in the two and three dimensional case. Let Ω ⊂ R(n = 2, 3) be a bounded domain where the boundary ∂Ω is C and let D be an open set satisfying D ⊂ Ω and Ω D is connected. The defect D consists of the union of disjoint bounded domains {Dj}j=1, where the boundary of D is Lipschitz continuous. First, we define the DN map for the magnetic Schrödinger equation with no defect D in Ω. Here, let D Au := ∑n j=1 DA,j(DA,ju), where DA,j := 1 i ∂j +Aj and A = (A1, A2, · · · , An). Definition 1. Suppose q ∈ L∞(Ω), q ≥ 0, A ∈ C(Ω, R). For a given f ∈ H(∂Ω), we say u ∈ H(Ω) is a weak solution to the following boundary value problem for the magnetic Schrödinger equation { D Au+ qu = 0 in Ω, u = f on ∂Ω, (1.1)
{"title":"Reconstruction of the Defect by the Enclosure Method for Inverse Problems of the Magnetic Schrödinger Operator","authors":"K. Kurata, Ryusei Yamashita","doi":"10.3836/tjm/1502179363","DOIUrl":"https://doi.org/10.3836/tjm/1502179363","url":null,"abstract":"This study is based on the paper [1]. We give the formula to extract the position and the shape of the defect D generated in the object (conductor) Ω from the observation data on the boundary ∂Ω for the magnetic Schrödinger operator by using the enclosure method proposed by Ikehata [2]. We show a reconstruction formula of the convex hull of the defect D from the observed data, assuming certain higher regularity for the potentials of the magnetic Schrödinger operator, under the Dirichlet condition or the Robin condition on the boundary ∂D in the two and three dimensional case. Let Ω ⊂ R(n = 2, 3) be a bounded domain where the boundary ∂Ω is C and let D be an open set satisfying D ⊂ Ω and Ω D is connected. The defect D consists of the union of disjoint bounded domains {Dj}j=1, where the boundary of D is Lipschitz continuous. First, we define the DN map for the magnetic Schrödinger equation with no defect D in Ω. Here, let D Au := ∑n j=1 DA,j(DA,ju), where DA,j := 1 i ∂j +Aj and A = (A1, A2, · · · , An). Definition 1. Suppose q ∈ L∞(Ω), q ≥ 0, A ∈ C(Ω, R). For a given f ∈ H(∂Ω), we say u ∈ H(Ω) is a weak solution to the following boundary value problem for the magnetic Schrödinger equation { D Au+ qu = 0 in Ω, u = f on ∂Ω, (1.1)","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46564749","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 discuss Monge-Ampère equations from the view point of differential geometry. It is known that a Monge–Ampère equation corresponds to a special exterior differential system on a 1-jet space. In this paper, we generalize Monge–Ampère equations and prove that a (k+ 1)st order generalized Monge–Ampère equation corresponds to a special exterior differential system on a k-jet space. Then its solution naturally corresponds to an integral manifold of the corresponding exterior differential system. Moreover, we verify that the Korteweg-de Vries (KdV) equation and the Cauchy–Riemann equations are examples of our equation. 2010 Mathematics Subject Classification. Primary 58A15; Secondary 58A17.
{"title":"On a Generalization of Monge–Ampère Equations and Monge–Ampère Systems","authors":"M. Kawamata, K. Shibuya","doi":"10.3836/tjm/1502179374","DOIUrl":"https://doi.org/10.3836/tjm/1502179374","url":null,"abstract":"We discuss Monge-Ampère equations from the view point of differential geometry. It is known that a Monge–Ampère equation corresponds to a special exterior differential system on a 1-jet space. In this paper, we generalize Monge–Ampère equations and prove that a (k+ 1)st order generalized Monge–Ampère equation corresponds to a special exterior differential system on a k-jet space. Then its solution naturally corresponds to an integral manifold of the corresponding exterior differential system. Moreover, we verify that the Korteweg-de Vries (KdV) equation and the Cauchy–Riemann equations are examples of our equation. 2010 Mathematics Subject Classification. Primary 58A15; Secondary 58A17.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47527732","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":"Virtual Knots with Properties of Kishino's Knot","authors":"Y. Ohyama, Migiwa Sakurai","doi":"10.3836/tjm/1502179365","DOIUrl":"https://doi.org/10.3836/tjm/1502179365","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43419315","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":"Predual of Weak Orlicz Spaces and its Applications to Fefferman-Stein Vector-valued Maximal Inequality","authors":"N. Hatano, Ryota Kawasumi, Takahiro Ono","doi":"10.3836/tjm/1502179373","DOIUrl":"https://doi.org/10.3836/tjm/1502179373","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42575822","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 Equivalence Classes of SO0(p,q)-actions on Sp+q−1 whose Restricted SO(p)×SO(q)-action is the Standard Action","authors":"T. Ono","doi":"10.3836/tjm/1502179370","DOIUrl":"https://doi.org/10.3836/tjm/1502179370","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42676081","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":"Preduals of Sobolev Multiplier Spaces for End Point Cases","authors":"Keng Hao Ooi","doi":"10.3836/tjm/1502179383","DOIUrl":"https://doi.org/10.3836/tjm/1502179383","url":null,"abstract":"","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43223187","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 Sobolev spaces with weights in the half-space $mathbb{R}^{N+1}_+={(x,y): x in mathbb{R}^N, y>0}$, adapted to the singular elliptic operators begin{equation*} mathcal L =y^{alpha_1}Delta_{x} +y^{alpha_2}left(D_{yy}+frac{c}{y}D_y -frac{b}{y^2}right). end{equation*}
研究了半空间$mathbb{R}^{N+1}_+={(x,y): x in mathbb{R}^N, y>0}$中具有权值的Sobolev空间，该空间适用于奇异椭圆算子 begin{equation*} mathcal L =y^{alpha_1}Delta_{x} +y^{alpha_2}left(D_{yy}+frac{c}{y}D_y -frac{b}{y^2}right). end{equation*}
{"title":"Anisotropic Sobolev Spaces with Weights","authors":"G. Metafune, L. Negro, C. Spina","doi":"10.3836/tjm/1502179386","DOIUrl":"https://doi.org/10.3836/tjm/1502179386","url":null,"abstract":"We study Sobolev spaces with weights in the half-space $mathbb{R}^{N+1}_+={(x,y): x in mathbb{R}^N, y>0}$, adapted to the singular elliptic operators begin{equation*} mathcal L =y^{alpha_1}Delta_{x} +y^{alpha_2}left(D_{yy}+frac{c}{y}D_y -frac{b}{y^2}right). end{equation*}","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46056074","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 generalize Bauer's Identical Congruence appearing in Hardy and Wright's book [6], Theorems 126 and 127. Bauer's Identical Congruence asserts that the polynomial $prod_t(x-t)$, where the product runs over a reduced residue system modulo a prime power $p^a$, is congruent (mod $p^a$) to the “simple” polynomial $(x^{p-1}-1)^{p^{a-1}}$ if $p>2$ and $(x^2-1)^{2^{a-2}}$ if $p=2$ and $ageqslant2$. Our article generalizes these results to a broader context, in which we find a “simple” form of the polynomial $prod_t(x-t)$, where the product runs over the solutions of the congruence $t^nequiv 1pmod{mathrm{P}^a}$ in the framework of the ring of algebraic integers of a given number field $mathbb{K}$, and where $mathrm{P}$ is a prime ideal.
{"title":"A Generalization of Bauer's Identical Congruence","authors":"Boaz Cohen","doi":"10.3836/tjm/1502179350","DOIUrl":"https://doi.org/10.3836/tjm/1502179350","url":null,"abstract":"In this paper we generalize Bauer's Identical Congruence appearing in Hardy and Wright's book [6], Theorems 126 and 127. Bauer's Identical Congruence asserts that the polynomial $prod_t(x-t)$, where the product runs over a reduced residue system modulo a prime power $p^a$, is congruent (mod $p^a$) to the “simple” polynomial $(x^{p-1}-1)^{p^{a-1}}$ if $p>2$ and $(x^2-1)^{2^{a-2}}$ if $p=2$ and $ageqslant2$. Our article generalizes these results to a broader context, in which we find a “simple” form of the polynomial $prod_t(x-t)$, where the product runs over the solutions of the congruence $t^nequiv 1pmod{mathrm{P}^a}$ in the framework of the ring of algebraic integers of a given number field $mathbb{K}$, and where $mathrm{P}$ is a prime ideal.","PeriodicalId":48976,"journal":{"name":"Tokyo Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45384480","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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