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Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases 非光滑情况下分数阶Dirichlet实现的Weyl渐近性
4区 数学
Mathematica Scandinavica Pub Date : 2023-10-26 DOI: 10.7146/math.scand.a-138002
Gerd Grubb
{"title":"Weyl asymptotics for fractional-order Dirichlet realizations in nonsmooth cases","authors":"Gerd Grubb","doi":"10.7146/math.scand.a-138002","DOIUrl":"https://doi.org/10.7146/math.scand.a-138002","url":null,"abstract":"Let $P$ be a symmetric $2a$-order classical strongly elliptic pseudodifferential operator with emph{even} symbol $p(x,xi)$ on $mathbb{R}^n $ ($0<a<1$), for example a perturbation of $(-Delta )^a$. Let $Omega subset mathbb{R}^n$ be bounded, and let $P_D$ be the Dirichlet realization in $L_2(Omega)$ defined under the exterior condition $u=0$ in $mathbb{R}^nsetminusOmega$. When $p(x,xi)$ and $Omega$ are $C^infty $, it is known that the eigenvalues $lambda_j$ (ordered in a nondecreasing sequence for $jto infty$) satisfy a Weyl asymptotic formula begin{equation*} lambda _j(P_{D})=C(P,Omega )j^{2a/n}+o(j^{2a/n}) text {for $jto infty $}, end{equation*} with $C(P,Omega)$ determined from the principal symbol of $P$. We now show that this result is valid for more general operators with a possibly nonsmooth $x$-dependence, over Lipschitz domains, and that it extends to $tilde P=P+P'+P”$, where $P'$ is an operator of order $<min{2a, a+frac 12}$ with certain mapping properties, and $P”$ is bounded in $L_2(Omega )$ (e.g. $P”=V(x)in L_infty(Omega)$). Also the regularity of eigenfunctions of $P_D$ is discussed.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134907331","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}
引用次数: 1
On scaling limits of random Halin-like maps 随机类halin地图的缩放极限
4区 数学
Mathematica Scandinavica Pub Date : 2023-10-26 DOI: 10.7146/math.scand.a-139930
Daniel Amankwah, Sigurdur Örn Stefánsson
{"title":"On scaling limits of random Halin-like maps","authors":"Daniel Amankwah, Sigurdur Örn Stefánsson","doi":"10.7146/math.scand.a-139930","DOIUrl":"https://doi.org/10.7146/math.scand.a-139930","url":null,"abstract":"We consider maps which are constructed from plane trees by assigning marks to the corners of each vertex and then connecting each pair of consecutive marks on their contour by a single edge. A measure is defined on the set of such maps by assigning Boltzmann weights to the faces. When every vertex has exactly one marked corner, these maps are dissections of a polygon which are bijectively related to non-crossing trees. When every vertex has at least one marked corner, the maps are outerplanar and each of its two-connected component is bijectively related to a non-crossing tree. We study the scaling limits of the maps under these conditions and establish that for certain choices of the weights the scaling limits are either the Brownian CRT or the α-stable looptrees of Curien and Kortchemski.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134907700","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}
引用次数: 1
Constructing stable vector bundles from curves with torsion normal bundle 从具有扭转法向束的曲线构造稳定向量束
IF 0.5 4区 数学
Mathematica Scandinavica Pub Date : 2023-06-05 DOI: 10.7146/math.scand.a-136533
Sergio Licanic
{"title":"Constructing stable vector bundles from curves with torsion normal bundle","authors":"Sergio Licanic","doi":"10.7146/math.scand.a-136533","DOIUrl":"https://doi.org/10.7146/math.scand.a-136533","url":null,"abstract":"Given a smooth irreducible curve $S$ with torsion normal bundle on a projective surface $X$, we provide a criterion for the non-emptiness of the moduli of slope stable vector bundles with prescribed Chern classes. The criterion is given in terms of the topology of the pair $(X,S)$.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41609429","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}
引用次数: 0
Binomial edge ideals over an exterior algebra 外代数上的二项式边理想
IF 0.5 4区 数学
Mathematica Scandinavica Pub Date : 2023-06-05 DOI: 10.7146/math.scand.a-137125
I. Peeva
{"title":"Binomial edge ideals over an exterior algebra","authors":"I. Peeva","doi":"10.7146/math.scand.a-137125","DOIUrl":"https://doi.org/10.7146/math.scand.a-137125","url":null,"abstract":"We introduce the study of binomial edge ideals over an exterior algebra.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43447085","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}
引用次数: 0
Vanishing Morrey integrability for Riesz potentials in Morrey-Orlicz spaces Morrey-Orlicz空间中Riesz势的消失Morrey可积性
IF 0.5 4区 数学
Mathematica Scandinavica Pub Date : 2023-06-05 DOI: 10.7146/math.scand.a-136539
Y. Mizuta, T. Shimomura
{"title":"Vanishing Morrey integrability for Riesz potentials in Morrey-Orlicz spaces","authors":"Y. Mizuta, T. Shimomura","doi":"10.7146/math.scand.a-136539","DOIUrl":"https://doi.org/10.7146/math.scand.a-136539","url":null,"abstract":"Our aim in this paper is to establish vanishing Morrey integrability for Riesz potentials of functions in Morrey-Orlicz spaces. We discuss the size of the exceptional sets by using a capacity and Hausdorff measure. We also give Trudinger-type exponential Morrey integrability for Riesz potentials of functions in Morrey-Orlicz spaces.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43563776","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}
引用次数: 0
Capacities from moduli in metric measure spaces 度量度量空间中模的容量
4区 数学
Mathematica Scandinavica Pub Date : 2023-06-05 DOI: 10.7146/math.scand.a-136662
Olli Martio
{"title":"Capacities from moduli in metric measure spaces","authors":"Olli Martio","doi":"10.7146/math.scand.a-136662","DOIUrl":"https://doi.org/10.7146/math.scand.a-136662","url":null,"abstract":"The concept of capacity is an indispensable tool for analysis and path families and their moduli play a fundamental role in a metric space $X$. It is shown that the $AM_p(Gamma )$- and $M_p(Gamma )$-modulus create the capacities, $mathrm {Cap}_p^{AM}(E,G)$ and $mathrm {Cap}_p^{M}(E,G)$, respectively, where Γ is the path family connecting an arbitrary set $E subset G$ to the complement of a bounded open set $G$. The capacities use Lipschitz functions and their upper gradients. For $p &gt; 1$ the capacities coincide but differ for $p=1$. For $p geq 1$ it is shown that the $mathrm {Cap}_p^{AM}(E,G)$-capacity equals to the classical variational Dirichlet capacity of the condenser $(E,G)$ and the $mathrm {Cap}_p^{M}(E,G)$-capacity to the $M_p(Gamma )$-modulus.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135657342","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}
引用次数: 0
Topologically stable and persistent points of group actions 群行为的拓扑稳定和持久点
IF 0.5 4区 数学
Mathematica Scandinavica Pub Date : 2023-02-20 DOI: 10.7146/math.scand.a-134098
A. G. Khan, T. Das
{"title":"Topologically stable and persistent points of group actions","authors":"A. G. Khan, T. Das","doi":"10.7146/math.scand.a-134098","DOIUrl":"https://doi.org/10.7146/math.scand.a-134098","url":null,"abstract":"In this paper, we introduce topologically stable points, persistent points, persistent property, persistent measures and almost persistent measures for first countable Hausdorff group actions of compact metric spaces. We prove that the set of all persistent points is measurable and it is closed if the action is equicontinuous. We also prove that the set of all persistent measures is a convex set and every almost persistent measure is a persistent measure. Finally, we prove that every equicontinuous pointwise topologically stable first countable Hausdorff group action of a compact metric space is persistent. In particular, every equicontinuous pointwise topologically stable flow is persistent.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41935957","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}
引用次数: 0
Generalized Bernstein functions 广义Bernstein函数
IF 0.5 4区 数学
Mathematica Scandinavica Pub Date : 2023-02-20 DOI: 10.7146/math.scand.a-134298
S. Koumandos, Henrik L. Pedersen
{"title":"Generalized Bernstein functions","authors":"S. Koumandos, Henrik L. Pedersen","doi":"10.7146/math.scand.a-134298","DOIUrl":"https://doi.org/10.7146/math.scand.a-134298","url":null,"abstract":"A class of functions called generalized Bernstein functions is studied. The fundamental properties of this class are given and its relation to generalized Stieltjes functions via the Laplace transform is investigated. The subclass of generalized Thorin-Bernstein functions is characterized in different ways. Examples of generalized Bernstein functions include incomplete gamma functions, Lerch's transcendent and some hypergeometric functions.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46301616","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}
引用次数: 2
On the regularity of small symbolic powers of edge ideals of graphs 关于图的边理想的小符号幂的正则性
IF 0.5 4区 数学
Mathematica Scandinavica Pub Date : 2023-02-20 DOI: 10.7146/math.scand.a-134104
S. A. Seyed Fakhari
{"title":"On the regularity of small symbolic powers of edge ideals of graphs","authors":"S. A. Seyed Fakhari","doi":"10.7146/math.scand.a-134104","DOIUrl":"https://doi.org/10.7146/math.scand.a-134104","url":null,"abstract":"Assume that $G$ is a graph with edge ideal $I(G)$ and let $I(G)^{(s)}$ denote the $s$-th symbolic power of $I(G)$. It is proved that for every integer $sgeq 1$, $$ mathrm{reg} (I(G)^{(s+1)})leq max bigl {mathrm{reg} (I(G))$$ $$+2s, mathrm{reg} bigl (I(G)^{(s+1)}+I(G)^sbigr )bigr }. $$ As a consequence, we conclude that $mathrm{reg} (I(G)^{(2)})leq mathrm{reg} (I(G))+2$, and $mathrm{reg} (I(G)^{(3)})leq mathrm{reg} (I(G))+4$. Moreover, it is shown that if for some integer $kgeq 1$, the graph $G$ has no odd cycle of length at most $2k-1$, then $mathrm{reg} (I(G)^{(s)})leq 2s+mathrm{reg} (I(G))-2$, for every integer $sleq k+1$. Finally, it is proven that $mathrm{reg} (I(G)^{(s)})=2s$, for $sin {2, 3, 4}$, provided that the complementary graph $overline {G}$ is chordal.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43424021","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}
引用次数: 1
The Haar state on the Vaksman-Soibelman quantum spheres Vaksman-Soibelman量子球上的哈尔态
IF 0.5 4区 数学
Mathematica Scandinavica Pub Date : 2022-12-21 DOI: 10.7146/math.scand.a-136693
Max Holst Mikkelsen, Jens Kaad
{"title":"The Haar state on the Vaksman-Soibelman quantum spheres","authors":"Max Holst Mikkelsen, Jens Kaad","doi":"10.7146/math.scand.a-136693","DOIUrl":"https://doi.org/10.7146/math.scand.a-136693","url":null,"abstract":"In this note we present explicit formulae for the Haar state on the Vaksman-Soibelman quantum spheres. Our formulae correct various statements appearing in the literature and our proof is straightforward relying simply on properties of the modular automorphism group for the Haar state.","PeriodicalId":49873,"journal":{"name":"Mathematica Scandinavica","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43862923","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}
引用次数: 2
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