Pub Date : 2022-12-04DOI: 10.7146/math.scand.a-133368
W. F. C. Barboza, H. D. de Lima, M. Velásquez
We deal with $n$-dimensional spacelike submanifolds immersed with parallel mean curvature vector (which is supposed to be either spacelike or timelike) in a pseudo-Riemannian space form $mathbb L_q^{n+p}(c)$ of index $1leq qleq p$ and constant sectional curvature $cin {-1,0,1}$. Under suitable constraints on the traceless second fundamental form, we adapt the technique developed by Yang and Li (Math. Notes 100 (2016) 298–308) to prove that such a spacelike submanifold must be totally umbilical. For this, we apply a maximum principle for complete noncompact Riemannian manifolds having polynomial volume growth, recently established by Alías, Caminha and Nascimento (Ann. Mat. Pura Appl. 200 (2021) 1637–1650).
{"title":"Gap type results for spacelike submanifolds with parallel mean curvature vector","authors":"W. F. C. Barboza, H. D. de Lima, M. Velásquez","doi":"10.7146/math.scand.a-133368","DOIUrl":"https://doi.org/10.7146/math.scand.a-133368","url":null,"abstract":"We deal with $n$-dimensional spacelike submanifolds immersed with parallel mean curvature vector (which is supposed to be either spacelike or timelike) in a pseudo-Riemannian space form $mathbb L_q^{n+p}(c)$ of index $1leq qleq p$ and constant sectional curvature $cin {-1,0,1}$. Under suitable constraints on the traceless second fundamental form, we adapt the technique developed by Yang and Li (Math. Notes 100 (2016) 298–308) to prove that such a spacelike submanifold must be totally umbilical. For this, we apply a maximum principle for complete noncompact Riemannian manifolds having polynomial volume growth, recently established by Alías, Caminha and Nascimento (Ann. Mat. Pura Appl. 200 (2021) 1637–1650).","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48956040","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 : 2022-11-10DOI: 10.7146/math.scand.a-136450
Giovanni Domenico Di Salvo
We present four approximation theorems for manifold–valued mappings. The first one approximates holomorphic embeddings on pseudoconvex domains in $mathbb{C}^n$ with holomorphic embeddings with dense images. The second theorem approximates holomorphic mappings on complex manifolds with bounded images with holomorphic mappings with dense images. The last two theorems work the other way around, constructing (in different settings) sequences of holomorphic mappings (embeddings in the first one) converging to a mapping with dense image defined on a given compact minus certain points (thus in general not holomorphic).
{"title":"Approximation and accumulation results of holomorphic mappings with dense image","authors":"Giovanni Domenico Di Salvo","doi":"10.7146/math.scand.a-136450","DOIUrl":"https://doi.org/10.7146/math.scand.a-136450","url":null,"abstract":"We present four approximation theorems for manifold–valued mappings. The first one approximates holomorphic embeddings on pseudoconvex domains in $mathbb{C}^n$ with holomorphic embeddings with dense images. The second theorem approximates holomorphic mappings on complex manifolds with bounded images with holomorphic mappings with dense images. The last two theorems work the other way around, constructing (in different settings) sequences of holomorphic mappings (embeddings in the first one) converging to a mapping with dense image defined on a given compact minus certain points (thus in general not holomorphic).","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49172508","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 : 2022-08-25DOI: 10.7146/math.scand.a-136656
Matthew Romney
We give an example of an Ahlfors $3$-regular, linearly locally connected metric space homeomorphic to $mathbb {R}^3$ containing a nondegenerate continuum $E$ with zero capacity, in the sense that the conformal modulus of the set of nontrivial curves intersecting $E$ is zero. We discuss this example in relation to the quasiconformal uniformization problem for metric spaces.
{"title":"Remarks on conformal modulus in metric spaces","authors":"Matthew Romney","doi":"10.7146/math.scand.a-136656","DOIUrl":"https://doi.org/10.7146/math.scand.a-136656","url":null,"abstract":"We give an example of an Ahlfors $3$-regular, linearly locally connected metric space homeomorphic to $mathbb {R}^3$ containing a nondegenerate continuum $E$ with zero capacity, in the sense that the conformal modulus of the set of nontrivial curves intersecting $E$ is zero. We discuss this example in relation to the quasiconformal uniformization problem for metric spaces.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48197046","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 : 2022-06-22DOI: 10.7146/math.scand.a-136741
Eduardo Scarparo
The AH conjecture relates the low-dimensional homology groups of a groupoid with the abelianization of its topological full group. We show that transformation groupoids of minimal actions of the infinite dihedral group on the Cantor set satisfy this conjecture. The proof uses Kakutani–Rokhlin partitions adapted to such systems.
{"title":"The AH conjecture for Cantor minimal dihedral systems","authors":"Eduardo Scarparo","doi":"10.7146/math.scand.a-136741","DOIUrl":"https://doi.org/10.7146/math.scand.a-136741","url":null,"abstract":"The AH conjecture relates the low-dimensional homology groups of a groupoid with the abelianization of its topological full group. We show that transformation groupoids of minimal actions of the infinite dihedral group on the Cantor set satisfy this conjecture. The proof uses Kakutani–Rokhlin partitions adapted to such systems.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45402761","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 : 2022-04-04DOI: 10.7146/math.scand.a-134458
D. Barlet
For a holomorphic function $f$ on a complex manifold $mathscr {M}$ we explain in this article that the distribution associated to $lvert frvert^{2alpha } (textrm{Log} lvert frvert^2)^q f^{-N}$ by taking the corresponding limit on the sets ${ lvert frvert geq varepsilon }$ when $varepsilon $ goes to $0$, coincides for $Re (alpha ) $ non negative and $q, N in mathbb {N}$, with the value at $lambda = alpha $ of the meromorphic extension of the distribution $lvert frvert^{2lambda } (textrm{Log} lvert frvert^2)^qf^{-N}$. This implies that any distribution in the $mathcal {D}_{mathscr {M}}$-module generated by such a distribution has the standard extension property. This implies a non $mathcal {O}_mathscr {M}$ torsion result for the $mathcal {D}_{mathscr {M}}$-module generated by such a distribution. As an application of this result we determine generators for the conjugate modules of the regular holonomic $mathcal {D}$-modules introduced and studied in [Barlet, D., On symmetric partial differential operators, Math. Z. 302 (2022), no. 3, 1627–1655] and [Barlet, D., On partial differential operators which annihilate the roots of the universal equation of degree $k$, arXiv:2101.01895] associated to the roots of universal equation of degree $k$, $z^k + sum _{h=1}^k (-1)^hsigma _hz^{k-h} = 0$.
对于复流形$mathscr｛M｝$上的全纯函数$f$，我们在本文中解释了与$lvert frvert ^｛2 alpha｝（textrm｛Log｝lvert f rvert ^2）^q f^｛-N｝$相关的分布，当$varepsilon$变为$0$时，通过对集合$｛lvert f rvert geqvarepsilion｝$取相应的极限，与$Re（alpha）$非负和$q，Ninmathbb｛N｝$重合，具有分布$lvert-frvert^｛2lambda｝（textrm｛Log｝lvert-f rvert^2）^qf^｛-N｝$的亚纯扩展的$lambda=alpha$处的值。这意味着$mathcal中的任何分布{D}_｛mathscr｛M｝｝$-由这样的分发生成的模块具有标准扩展属性。这意味着非$mathcal{O}_$mathcal的mathscr｛M｝$扭转结果{D}_｛mathscr｛M｝｝$-由这样的分发生成的模块。作为这一结果的一个应用，我们确定了在[Ballet，D.，On symmetric偏微分算子，Math.Z.302（2022），no.31627-1655]和[Balet，D.，On-P偏微分算子的共轭模的生成元，这些共轭模在[Balllet，D.中引入并研究了正则完整$mathcal｛D}-模，arXiv:21011895]与普遍次方程$k$的根有关，$z^k+sum_{h=1}^k（-1）^hsigmahz^{k-h}=0$。
{"title":"On principal value and standard extension of distributions","authors":"D. Barlet","doi":"10.7146/math.scand.a-134458","DOIUrl":"https://doi.org/10.7146/math.scand.a-134458","url":null,"abstract":"For a holomorphic function $f$ on a complex manifold $mathscr {M}$ we explain in this article that the distribution associated to $lvert frvert^{2alpha } (textrm{Log} lvert frvert^2)^q f^{-N}$ by taking the corresponding limit on the sets ${ lvert frvert geq varepsilon }$ when $varepsilon $ goes to $0$, coincides for $Re (alpha ) $ non negative and $q, N in mathbb {N}$, with the value at $lambda = alpha $ of the meromorphic extension of the distribution $lvert frvert^{2lambda } (textrm{Log} lvert frvert^2)^qf^{-N}$. This implies that any distribution in the $mathcal {D}_{mathscr {M}}$-module generated by such a distribution has the standard extension property. This implies a non $mathcal {O}_mathscr {M}$ torsion result for the $mathcal {D}_{mathscr {M}}$-module generated by such a distribution. As an application of this result we determine generators for the conjugate modules of the regular holonomic $mathcal {D}$-modules introduced and studied in [Barlet, D., On symmetric partial differential operators, Math. Z. 302 (2022), no. 3, 1627–1655] and [Barlet, D., On partial differential operators which annihilate the roots of the universal equation of degree $k$, arXiv:2101.01895] associated to the roots of universal equation of degree $k$, $z^k + sum _{h=1}^k (-1)^hsigma _hz^{k-h} = 0$.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43574855","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 : 2022-02-24DOI: 10.7146/math.scand.a-129287
Gereon Quick, Therese Strand, G. Wilson
We study which quadratic forms are representable as the local degree of a map $f colon mathbb{A}^n to mathbb{A}^n$ with an isolated zero at $0$, following the work of Kass and Wickelgren who established the connection to the quadratic form of Eisenbud, Khimshiashvili, and Levine. Our main observation is that over some base fields $k$, not all quadratic forms are representable as a local degree. Empirically the local degree of a map $f colon mathbb{A}^n to mathbb{A}^n$ has many hyperbolic summands, and we prove that in fact this is the case for local degrees of low rank. We establish a complete classification of the quadratic forms of rank at most $7$ that are representable as the local degree of a map over all base fields of characteristic different from $2$. The number of hyperbolic summands was also studied by Eisenbud and Levine, where they establish general bounds on the number of hyperbolic forms that must appear in a quadratic form that is representable as a local degree. Our proof method is elementary and constructive in the case of rank 5 local degrees, while the work of Eisenbud and Levine is more general. We provide further families of examples that verify that the bounds of Eisenbud and Levine are tight in several cases.
{"title":"Representability of the local motivic Brouwer degree","authors":"Gereon Quick, Therese Strand, G. Wilson","doi":"10.7146/math.scand.a-129287","DOIUrl":"https://doi.org/10.7146/math.scand.a-129287","url":null,"abstract":"We study which quadratic forms are representable as the local degree of a map $f colon mathbb{A}^n to mathbb{A}^n$ with an isolated zero at $0$, following the work of Kass and Wickelgren who established the connection to the quadratic form of Eisenbud, Khimshiashvili, and Levine. Our main observation is that over some base fields $k$, not all quadratic forms are representable as a local degree. Empirically the local degree of a map $f colon mathbb{A}^n to mathbb{A}^n$ has many hyperbolic summands, and we prove that in fact this is the case for local degrees of low rank. We establish a complete classification of the quadratic forms of rank at most $7$ that are representable as the local degree of a map over all base fields of characteristic different from $2$. The number of hyperbolic summands was also studied by Eisenbud and Levine, where they establish general bounds on the number of hyperbolic forms that must appear in a quadratic form that is representable as a local degree. Our proof method is elementary and constructive in the case of rank 5 local degrees, while the work of Eisenbud and Levine is more general. We provide further families of examples that verify that the bounds of Eisenbud and Levine are tight in several cases.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44581742","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 : 2022-02-24DOI: 10.7146/math.scand.a-135820
Bjørn Jensen, K. Knudsen, Hjordis Schluter
In acousto-electric tomography, the objective is to extract information about the interior electrical conductivity in a physical body from knowledge of the interior power density data generated from prescribed boundary conditions for the governing elliptic partial differential equation. In this note, we consider the problem when the controlled boundary conditions are applied only on a small subset of the full boundary. We demonstrate using the unique continuation principle that the Runge approximation property is valid also for this special case of limited view data. As a consequence, we guarantee the existence of finitely many boundary conditions such that the corresponding solutions locally satisfy a non-vanishing gradient condition. This condition is essential for conductivity reconstruction from power density data. In addition, we adapt an existing reconstruction method intended for the full data situation to our setting. We implement the method numerically and investigate the opportunities and shortcomings when reconstructing from two fixed boundary conditions.
{"title":"Conductivity reconstruction from power density data in limited view","authors":"Bjørn Jensen, K. Knudsen, Hjordis Schluter","doi":"10.7146/math.scand.a-135820","DOIUrl":"https://doi.org/10.7146/math.scand.a-135820","url":null,"abstract":"In acousto-electric tomography, the objective is to extract information about the interior electrical conductivity in a physical body from knowledge of the interior power density data generated from prescribed boundary conditions for the governing elliptic partial differential equation. In this note, we consider the problem when the controlled boundary conditions are applied only on a small subset of the full boundary. We demonstrate using the unique continuation principle that the Runge approximation property is valid also for this special case of limited view data. As a consequence, we guarantee the existence of finitely many boundary conditions such that the corresponding solutions locally satisfy a non-vanishing gradient condition. This condition is essential for conductivity reconstruction from power density data. In addition, we adapt an existing reconstruction method intended for the full data situation to our setting. We implement the method numerically and investigate the opportunities and shortcomings when reconstructing from two fixed boundary conditions.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47985554","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 : 2022-02-24DOI: 10.7146/math.scand.a-129245
T. Ohno, T. Shimomura
Our aim in this paper is to give Trudinger-type inequalities for Riesz potentials of functions in Musielak-Orlicz-Morrey spaces of an integral form over non-doubling metric measure spaces. Our result are new even for the doubling metric measure setting. As a corollary, we give Trudinger-type inequalities on Musielak-Orlicz-Morrey spaces of an integral form in the framework of double phase functions with variable exponents.
{"title":"Trudinger-type inequalities on Musielak-Orlicz-Morrey spaces of an integral form","authors":"T. Ohno, T. Shimomura","doi":"10.7146/math.scand.a-129245","DOIUrl":"https://doi.org/10.7146/math.scand.a-129245","url":null,"abstract":"Our aim in this paper is to give Trudinger-type inequalities for Riesz potentials of functions in Musielak-Orlicz-Morrey spaces of an integral form over non-doubling metric measure spaces. Our result are new even for the doubling metric measure setting. As a corollary, we give Trudinger-type inequalities on Musielak-Orlicz-Morrey spaces of an integral form in the framework of double phase functions with variable exponents.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42603164","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 : 2022-02-21DOI: 10.7146/math.scand.a-136634
D. Lu
We introduce the notion of quasi-linearity and prove it is necessary for a monomial ideal to have a linear resolution and clarify all the quasi-linear monomial ideals generated in degree $2$. We also introduce the notion of a strongly linear monomial over a monomial ideal and prove that if $mathbf {u}$ is a monomial strongly linear over $I$ then $I$ has a linear resolution (respectively is quasi-linear) if and only if $I+mathbf {u}mathfrak {p}$ has a linear resolution (respectively is quasi-linear). Here $mathfrak {p}$ is any monomial prime ideal.
{"title":"Linear resolutions and quasi-linearity of monomial ideals","authors":"D. Lu","doi":"10.7146/math.scand.a-136634","DOIUrl":"https://doi.org/10.7146/math.scand.a-136634","url":null,"abstract":"We introduce the notion of quasi-linearity and prove it is necessary for a monomial ideal to have a linear resolution and clarify all the quasi-linear monomial ideals generated in degree $2$. We also introduce the notion of a strongly linear monomial over a monomial ideal and prove that if $mathbf {u}$ is a monomial strongly linear over $I$ then $I$ has a linear resolution (respectively is quasi-linear) if and only if $I+mathbf {u}mathfrak {p}$ has a linear resolution (respectively is quasi-linear). Here $mathfrak {p}$ is any monomial prime ideal.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49566635","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 : 2022-01-04DOI: 10.7146/math.scand.a-136499
Souvik Dey, D. Ghosh
We analyze whether Ulrich modules, not necessarily maximal CM (Cohen-Macaulay), can be used as test modules, which detect finite homological dimensions of modules. We prove that Ulrich modules over CM local rings have maximal complexity and curvature. Various new characterizations of local rings are provided in terms of Ulrich modules. We show that every Ulrich module of dimension $s$ over a local ring is $(s+1)$-Tor-rigid-test, but not $s$−Tor-rigid in general (where $sge 1$). Over a deformation of a CM local ring of minimal multiplicity, we also study Tor rigidity.
{"title":"Complexity and rigidity of Ulrich modules, and some applications","authors":"Souvik Dey, D. Ghosh","doi":"10.7146/math.scand.a-136499","DOIUrl":"https://doi.org/10.7146/math.scand.a-136499","url":null,"abstract":"We analyze whether Ulrich modules, not necessarily maximal CM (Cohen-Macaulay), can be used as test modules, which detect finite homological dimensions of modules. We prove that Ulrich modules over CM local rings have maximal complexity and curvature. Various new characterizations of local rings are provided in terms of Ulrich modules. We show that every Ulrich module of dimension $s$ over a local ring is $(s+1)$-Tor-rigid-test, but not $s$−Tor-rigid in general (where $sge 1$). Over a deformation of a CM local ring of minimal multiplicity, we also study Tor rigidity.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45588373","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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