Pub Date : 2019-04-07DOI: 10.7146/math.scand.a-119382
P. Sahandi, Tirdad Sharif, S. Yassemi
We introduce new homological dimensions, namely the Cohen-Macaulay projective, injective and flat dimensions for homologically bounded complexes. Among other things we show that (a) these invariants characterize the Cohen-Macaulay property for local rings, (b) Cohen-Macaulay flat dimension fits between the Gorenstein flat dimension and the large restricted flat dimension, and (c) Cohen-Macaulay injective dimension fits between the Gorenstein injective dimension and the Chouinard invariant.
{"title":"Cohen-Macaulay homological dimensions","authors":"P. Sahandi, Tirdad Sharif, S. Yassemi","doi":"10.7146/math.scand.a-119382","DOIUrl":"https://doi.org/10.7146/math.scand.a-119382","url":null,"abstract":"We introduce new homological dimensions, namely the Cohen-Macaulay projective, injective and flat dimensions for homologically bounded complexes. Among other things we show that (a) these invariants characterize the Cohen-Macaulay property for local rings, (b) Cohen-Macaulay flat dimension fits between the Gorenstein flat dimension and the large restricted flat dimension, and (c) Cohen-Macaulay injective dimension fits between the Gorenstein injective dimension and the Chouinard invariant.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41625570","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 : 2019-04-07DOI: 10.7146/MATH.SCAND.A-121016
L. Robert, Luis Santiago
We revise the construction of the augmented Cuntz semigroup functor used by the first author to classify inductive limits of $1$-dimensional noncommutative CW complexes. The original construction has good functorial properties when restricted to the class of C*-algebras of stable rank one. The construction proposed here has good properties for all C*-algebras: we show that the augmented Cuntz semigroup is a stable, continuous, split exact functor, from the category of C*-algebras to the category of Cu-semigroups.
{"title":"A revised augmented Cuntz semigroup","authors":"L. Robert, Luis Santiago","doi":"10.7146/MATH.SCAND.A-121016","DOIUrl":"https://doi.org/10.7146/MATH.SCAND.A-121016","url":null,"abstract":"We revise the construction of the augmented Cuntz semigroup functor used by the first author to classify inductive limits of $1$-dimensional noncommutative CW complexes. The original construction has good functorial properties when restricted to the class of C*-algebras of stable rank one. The construction proposed here has good properties for all C*-algebras: we show that the augmented Cuntz semigroup is a stable, continuous, split exact functor, from the category of C*-algebras to the category of Cu-semigroups.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47024793","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 : 2019-01-13DOI: 10.7146/MATH.SCAND.A-109674
M. Stoll
In the paper we characterize the reproducing kernel $mathcal {K}_{n,h}$ for the Hardy space $mathcal {H}^2$ of hyperbolic harmonic functions on the unit ball $mathbb {B}$ in $mathbb {R}^n$. Specifically we prove that [ mathcal {K}_{n,h}(x,y) = sum _{alpha =0}^infty S_{n,alpha }(lvert xrvert )S_{n,alpha }(lvert yrvert ) Z_alpha (x,y), ] where the series converges absolutely and uniformly on $Ktimes mathbb {B}$ for every compact subset $K$ of $mathbb {B}$. In the above, $S_{n,alpha }$ is a hypergeometric function and $Z_alpha $ is the reproducing kernel of the space of spherical harmonics of degree α. In the paper we prove that [ 0le mathcal K_{n,h}(x,y) le frac {C_n}{(1-2langle x,yrangle + lvert x rvert^2 lvert y rvert^2)^{n-1}}, ] where $C_n$ is a constant depending only on $n$. It is known that the diagonal function $mathcal K_{n,h}(x,x)$ is a radial eigenfunction of the hyperbolic Laplacian $varDelta_h $ on $mathbb{B} $ with eigenvalue $lambda _2 = 8(n-1)^2$. The result for $n=4$ provides motivation that leads to an explicit characterization of all radial eigenfunctions of $varDelta_h $ on $mathbb{B} $. Specifically, if $g$ is a radial eigenfunction of $varDelta_h $ with eigenvalue $lambda _alpha = 4(n-1)^2alpha (alpha -1)$, then [ g(r) = g(0) frac {p_{n,alpha }(r^2)}{(1-r^2)^{(alpha -1)(n-1)}}, ] where $p_{n,alpha }$ is again a hypergeometric function. If α is an integer, then $p_{n,alpha }(r^2)$ is a polynomial of degree $2(alpha -1)(n-1)$.
{"title":"The reproducing kernel of $mathcal H^2$ and radial eigenfunctions of the hyperbolic Laplacian","authors":"M. Stoll","doi":"10.7146/MATH.SCAND.A-109674","DOIUrl":"https://doi.org/10.7146/MATH.SCAND.A-109674","url":null,"abstract":"In the paper we characterize the reproducing kernel $mathcal {K}_{n,h}$ for the Hardy space $mathcal {H}^2$ of hyperbolic harmonic functions on the unit ball $mathbb {B}$ in $mathbb {R}^n$. Specifically we prove that [ mathcal {K}_{n,h}(x,y) = sum _{alpha =0}^infty S_{n,alpha }(lvert xrvert )S_{n,alpha }(lvert yrvert ) Z_alpha (x,y), ] where the series converges absolutely and uniformly on $Ktimes mathbb {B}$ for every compact subset $K$ of $mathbb {B}$. In the above, $S_{n,alpha }$ is a hypergeometric function and $Z_alpha $ is the reproducing kernel of the space of spherical harmonics of degree α. In the paper we prove that [ 0le mathcal K_{n,h}(x,y) le frac {C_n}{(1-2langle x,yrangle + lvert x rvert^2 lvert y rvert^2)^{n-1}}, ] where $C_n$ is a constant depending only on $n$. It is known that the diagonal function $mathcal K_{n,h}(x,x)$ is a radial eigenfunction of the hyperbolic Laplacian $varDelta_h $ on $mathbb{B} $ with eigenvalue $lambda _2 = 8(n-1)^2$. The result for $n=4$ provides motivation that leads to an explicit characterization of all radial eigenfunctions of $varDelta_h $ on $mathbb{B} $. Specifically, if $g$ is a radial eigenfunction of $varDelta_h $ with eigenvalue $lambda _alpha = 4(n-1)^2alpha (alpha -1)$, then [ g(r) = g(0) frac {p_{n,alpha }(r^2)}{(1-r^2)^{(alpha -1)(n-1)}}, ] where $p_{n,alpha }$ is again a hypergeometric function. If α is an integer, then $p_{n,alpha }(r^2)$ is a polynomial of degree $2(alpha -1)(n-1)$.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47396391","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 : 2019-01-13DOI: 10.7146/MATH.SCAND.A-109996
C. Novelli
We describe the Kleiman-Mori cones of Fano manifolds of large pseudoindex that admit a structure of smooth blow-up.
我们描述了具有大伪指数的Fano流形的Kleiman-Mori锥，它允许光滑爆破的结构。
{"title":"Blow-ups and Fano manifolds of large pseudoindex","authors":"C. Novelli","doi":"10.7146/MATH.SCAND.A-109996","DOIUrl":"https://doi.org/10.7146/MATH.SCAND.A-109996","url":null,"abstract":"We describe the Kleiman-Mori cones of Fano manifolds of large pseudoindex that admit a structure of smooth blow-up.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48322631","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 : 2019-01-13DOI: 10.7146/MATH.SCAND.A-105504
L. Grafakos, P. Mohanty, Saurabh Shrivastava
In this paper we prove weighted estimates for a class of smooth multilinear square functions with respect to multilinear $A_{vec P}$ weights. In particular, we establish weighted estimates for the smooth multilinear square functions associated with disjoint cubes of equivalent side-lengths. As a consequence, for this particular class of multilinear square functions, we provide an affirmative answer to a question raised by Benea and Bernicot (Forum Math. Sigma 4, 2016, e26) about unweighted estimates for smooth bilinear square functions.
{"title":"Multilinear square functions and multiple weights","authors":"L. Grafakos, P. Mohanty, Saurabh Shrivastava","doi":"10.7146/MATH.SCAND.A-105504","DOIUrl":"https://doi.org/10.7146/MATH.SCAND.A-105504","url":null,"abstract":"In this paper we prove weighted estimates for a class of smooth multilinear square functions with respect to multilinear $A_{vec P}$ weights. In particular, we establish weighted estimates for the smooth multilinear square functions associated with disjoint cubes of equivalent side-lengths. As a consequence, for this particular class of multilinear square functions, we provide an affirmative answer to a question raised by Benea and Bernicot (Forum Math. Sigma 4, 2016, e26) about unweighted estimates for smooth bilinear square functions.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2019-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48703267","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 : 2018-12-18DOI: 10.7146/MATH.SCAND.A-109985
Arijit Ganguly, Anish Ghosh
We study some problems in metric Diophantine approximation over local fields of positive characteristic.
研究了局部正特征域上度量丢番图近似的几个问题。
{"title":"Remarks on Diophantine approximation in function fields","authors":"Arijit Ganguly, Anish Ghosh","doi":"10.7146/MATH.SCAND.A-109985","DOIUrl":"https://doi.org/10.7146/MATH.SCAND.A-109985","url":null,"abstract":"We study some problems in metric Diophantine approximation over local fields of positive characteristic.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2018-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41981442","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 : 2018-12-16DOI: 10.7146/math.scand.a-119708
P. Åhag, R. Czyż, Lisa Hed
We characterize those compact sets for which the Dirichlet problem has a solution within the class of continuous $m$-subharmonic functions defined on a compact set, and then within the class of $m$-harmonic functions.
{"title":"The Dirichlet problem for $m$-subharmonic functions on compact sets","authors":"P. Åhag, R. Czyż, Lisa Hed","doi":"10.7146/math.scand.a-119708","DOIUrl":"https://doi.org/10.7146/math.scand.a-119708","url":null,"abstract":"We characterize those compact sets for which the Dirichlet problem has a solution within the class of continuous $m$-subharmonic functions defined on a compact set, and then within the class of $m$-harmonic functions.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2018-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47249935","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 : 2018-12-13DOI: 10.7146/math.scand.a-117570
S. Frick, K. Petersen, Sandi Shields
We develop conditions for the coding of a Bratteli-Vershik system according to initial path segments to be periodic, equivalently for a constructive symbolic recursive scheme corresponding to a cutting and stacking process to produce a periodic sequence. This is a step toward understanding when a Bratteli-Vershik system can be essentially faithfully represented by means of a natural coding as a subshift on a finite alphabet.
{"title":"Periodic codings of Bratteli-Vershik systems","authors":"S. Frick, K. Petersen, Sandi Shields","doi":"10.7146/math.scand.a-117570","DOIUrl":"https://doi.org/10.7146/math.scand.a-117570","url":null,"abstract":"We develop conditions for the coding of a Bratteli-Vershik system according to initial path segments to be periodic, equivalently for a constructive symbolic recursive scheme corresponding to a cutting and stacking process to produce a periodic sequence. This is a step toward understanding when a Bratteli-Vershik system can be essentially faithfully represented by means of a natural coding as a subshift on a finite alphabet.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2018-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45194569","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 : 2018-11-29DOI: 10.7146/math.scand.a-120488
D. Sprehn, Nathalie Wahl
We give an overview of the general framework of forms of Bak, Tits and Wall, when restricting to vector spaces over fields, and describe its relationship to the classical notions of Hermitian, alternating and quadratic forms. We then prove a version of Witt's lemma in this context, showing in particular that the action of the group of isometries of a space equipped with a form is transitive on isometric subspaces.
{"title":"Forms over fields and Witt's lemma","authors":"D. Sprehn, Nathalie Wahl","doi":"10.7146/math.scand.a-120488","DOIUrl":"https://doi.org/10.7146/math.scand.a-120488","url":null,"abstract":"We give an overview of the general framework of forms of Bak, Tits and Wall, when restricting to vector spaces over fields, and describe its relationship to the classical notions of Hermitian, alternating and quadratic forms. We then prove a version of Witt's lemma in this context, showing in particular that the action of the group of isometries of a space equipped with a form is transitive on isometric subspaces.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2018-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47813950","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 : 2018-10-30DOI: 10.7146/math.scand.a-119684
G. Botelho, Leodan A. Torres
We generalize the classical notion of adjoint of a linear operator and the Aron-Schottenloher notion of adjoint of a homogeneous polynomial. The general (nonlinear) notion is shown to enjoy several properties enjoyed by the classical (linear) ones, nevertheless new interesting phenomena arise in the nonlinear theory. The proofs are not always simple adaptations of the linear cases, actually nonlinear arguments are often required. Applications of the generalized adjoints to Lindström-Schlüchtermann type theorems for composition operators are provided.
{"title":"Generalized adjoints and applications to composition operators","authors":"G. Botelho, Leodan A. Torres","doi":"10.7146/math.scand.a-119684","DOIUrl":"https://doi.org/10.7146/math.scand.a-119684","url":null,"abstract":"We generalize the classical notion of adjoint of a linear operator and the Aron-Schottenloher notion of adjoint of a homogeneous polynomial. The general (nonlinear) notion is shown to enjoy several properties enjoyed by the classical (linear) ones, nevertheless new interesting phenomena arise in the nonlinear theory. The proofs are not always simple adaptations of the linear cases, actually nonlinear arguments are often required. Applications of the generalized adjoints to Lindström-Schlüchtermann type theorems for composition operators are provided.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2018-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47722003","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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