We study sufficient conditions under which a nowhere scattered $mathrm {C}^*$-algebra $A$ has a nowhere scattered multiplier algebra $mathcal {M}(A)$, that is, we study when $mathcal {M}(A)$ has no nonzero, elementary ideal-quotients. In particular, we prove that a $sigma$-unital $mathrm {C}^*$-algebra $A$ of
(i) finite nuclear dimension, or
(ii) real rank zero, or
(iii) stable rank one with $k$-comparison,
is nowhere scattered if and only if $mathcal {M}(A)$
{"title":"Nowhere scattered multiplier algebras","authors":"Eduard Vilalta","doi":"10.1017/prm.2023.123","DOIUrl":"https://doi.org/10.1017/prm.2023.123","url":null,"abstract":"<p>We study sufficient conditions under which a nowhere scattered <span><span><span data-mathjax-type=\"texmath\"><span>$mathrm {C}^*$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104121729565-0175:S0308210523001233:S0308210523001233_inline1.png\"/></span></span>-algebra <span><span><span data-mathjax-type=\"texmath\"><span>$A$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104121729565-0175:S0308210523001233:S0308210523001233_inline2.png\"/></span></span> has a nowhere scattered multiplier algebra <span><span><span data-mathjax-type=\"texmath\"><span>$mathcal {M}(A)$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104121729565-0175:S0308210523001233:S0308210523001233_inline3.png\"/></span></span>, that is, we study when <span><span><span data-mathjax-type=\"texmath\"><span>$mathcal {M}(A)$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104121729565-0175:S0308210523001233:S0308210523001233_inline4.png\"/></span></span> has no nonzero, elementary ideal-quotients. In particular, we prove that a <span><span><span data-mathjax-type=\"texmath\"><span>$sigma$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104121729565-0175:S0308210523001233:S0308210523001233_inline5.png\"/></span></span>-unital <span><span><span data-mathjax-type=\"texmath\"><span>$mathrm {C}^*$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104121729565-0175:S0308210523001233:S0308210523001233_inline6.png\"/></span></span>-algebra <span><span><span data-mathjax-type=\"texmath\"><span>$A$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104121729565-0175:S0308210523001233:S0308210523001233_inline7.png\"/></span></span> of</p><ol><li><p><span>(i)</span> finite nuclear dimension, or</p></li><li><p><span>(ii)</span> real rank zero, or</p></li><li><p><span>(iii)</span> stable rank one with <span><span><span data-mathjax-type=\"texmath\"><span>$k$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104121729565-0175:S0308210523001233:S0308210523001233_inline8.png\"/></span></span>-comparison,</p></li></ol> is nowhere scattered if and only if <span><span><span data-mathjax-type=\"texmath\"><span>$mathcal {M}(A)$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240104121729565-0175:S0308210523001233:S0308210523001233","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139103518","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider Calderón's problem for the connection Laplacian on a real-analytic vector bundle over a manifold with boundary. We prove a uniqueness result for this problem when all geometric data are real-analytic, recovering the topology and geometry of a vector bundle up to a gauge transformation and an isometry of the base manifold.
{"title":"On Calderon's problem for the connection Laplacian","authors":"Ravil Gabdurakhmanov, Gerasim Kokarev","doi":"10.1017/prm.2023.127","DOIUrl":"https://doi.org/10.1017/prm.2023.127","url":null,"abstract":"<p>We consider Calderón's problem for the connection Laplacian on a real-analytic vector bundle over a manifold with boundary. We prove a uniqueness result for this problem when all geometric data are real-analytic, recovering the topology and geometry of a vector bundle up to a gauge transformation and an isometry of the base manifold.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"17 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Koushik Brahma, Bikramaditya Naskar, Soumen Sarkar, Subhankar Sau
In this paper, we compute the LS-category and equivariant LS-category of a small cover and its real moment angle manifold. We calculate a tight lower bound for the topological complexity of many small covers over a product of simplices. Then we compute symmetric topological complexity of several small covers over a product of simplices. We calculate the LS one-category of real Bott manifolds and infinitely many small covers.
在本文中，我们计算了小盖及其实矩角流形的 LS 类别和等变 LS 类别。我们计算了简约积上许多小盖的拓扑复杂性的严格下限。然后，我们计算简约积上多个小盖的对称拓扑复杂性。我们计算了实底流形和无限多小盖的 LS 单类别。
{"title":"Various topological complexities of small covers and real Bott manifolds","authors":"Koushik Brahma, Bikramaditya Naskar, Soumen Sarkar, Subhankar Sau","doi":"10.1017/prm.2023.124","DOIUrl":"https://doi.org/10.1017/prm.2023.124","url":null,"abstract":"In this paper, we compute the LS-category and equivariant LS-category of a small cover and its real moment angle manifold. We calculate a tight lower bound for the topological complexity of many small covers over a product of simplices. Then we compute symmetric topological complexity of several small covers over a product of simplices. We calculate the LS one-category of real Bott manifolds and infinitely many small covers.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"23 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139051416","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The present paper deals with the non-real eigenvalues for singular indefinite Sturm–Liouville problems. The lower bounds on non-real eigenvalues for this singular problem associated with a special separated boundary condition are obtained.
{"title":"The lower bounds of non-real eigenvalues for singular indefinite Sturm–Liouville problems","authors":"Fu Sun","doi":"10.1017/prm.2023.126","DOIUrl":"https://doi.org/10.1017/prm.2023.126","url":null,"abstract":"The present paper deals with the non-real eigenvalues for singular indefinite Sturm–Liouville problems. The lower bounds on non-real eigenvalues for this singular problem associated with a special separated boundary condition are obtained.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"10 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139027616","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $Omega subset mathbb {R}^N$ ($Ngeq 3$) be a $C^2$ bounded domain and $Sigma subset partial Omega$ be a $C^2$ compact submanifold without boundary, of dimension $k$, $0leq k leq N-1$. We assume that $Sigma = {0}$ if $k = 0$ and $Sigma =partial Omega$ if
We give a new proof of rationality of stable commutator length (scl) of certain elements in surface groups: those represented by curves that do not fill the surface. Such elements always admit extremal surfaces for scl. These results also hold more generally for non-filling $1$–chains.
{"title":"On stable commutator length of non-filling curves in surfaces","authors":"Max Forester, Justin Malestein","doi":"10.1017/prm.2023.121","DOIUrl":"https://doi.org/10.1017/prm.2023.121","url":null,"abstract":"<p>We give a new proof of rationality of stable commutator length (scl) of certain elements in surface groups: those represented by curves that do not fill the surface. Such elements always admit extremal surfaces for scl. These results also hold more generally for non-filling <span><span><span data-mathjax-type=\"texmath\"><span>$1$</span></span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20231213131917887-0965:S030821052300121X:S030821052300121X_inline1.png\"/></span></span>–chains.</p>","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"287 1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138630914","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Noting a curious link between Andrews’ even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is twofold. Firstly, we derive results for certain restricted partitions with even parts below odd parts. These include a Franklin-type involution proving a parametrized identity that generalizes Andrews’ bivariate generating function, and two families of Andrews–Beck type congruences. Secondly, we introduce several new subsets of partitions that are stable (i.e. invariant under conjugation) and explore their connections with three third-order mock theta functions $omega (q)$, $nu (q)$, and $psi ^{(3)}(q)$, introduced by Ramanujan and Watson.
{"title":"Partitions with parts separated by parity: conjugation, congruences and the mock theta functions","authors":"Shishuo Fu, Dazhao Tang","doi":"10.1017/prm.2023.119","DOIUrl":"https://doi.org/10.1017/prm.2023.119","url":null,"abstract":"Noting a curious link between Andrews’ even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is twofold. Firstly, we derive results for certain restricted partitions with even parts below odd parts. These include a Franklin-type involution proving a parametrized identity that generalizes Andrews’ bivariate generating function, and two families of Andrews–Beck type congruences. Secondly, we introduce several new subsets of partitions that are stable (i.e. invariant under conjugation) and explore their connections with three third-order mock theta functions <jats:inline-formula> <jats:alternatives> <jats:tex-math>$omega (q)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210523001191_inline1.png\" /> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:tex-math>$nu (q)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210523001191_inline2.png\" /> </jats:alternatives> </jats:inline-formula>, and <jats:inline-formula> <jats:alternatives> <jats:tex-math>$psi ^{(3)}(q)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210523001191_inline3.png\" /> </jats:alternatives> </jats:inline-formula>, introduced by Ramanujan and Watson.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":" 23","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The asymptotic mean value Laplacian—AMV Laplacian—extends the Laplace operator from $mathbb {R}^n$ to metric measure spaces through limits of averaging integrals. The AMV Laplacian is however not a symmetric operator in general. Therefore, we consider a symmetric version of the AMV Laplacian, and focus lies on when the symmetric and non-symmetric AMV Laplacians coincide. Besides Riemannian and 3D contact sub-Riemannian manifolds, we show that they are identical on a large class of metric measure spaces, including locally Ahlfors regular spaces with suitably vanishing distortion. In addition, we study the context of weighted domains of $mathbb {R}^n$—where the two operators typically differ—and provide explicit formulae for these operators, including points where the weight vanishes.
{"title":"Symmetrized and non-symmetrizedasymptotic mean value Laplacian in metric measure spaces","authors":"Andreas Minne, David Tewodrose","doi":"10.1017/prm.2023.118","DOIUrl":"https://doi.org/10.1017/prm.2023.118","url":null,"abstract":"The asymptotic mean value Laplacian—AMV Laplacian—extends the Laplace operator from <jats:inline-formula> <jats:alternatives> <jats:tex-math>$mathbb {R}^n$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030821052300118X_inline1.png\" /> </jats:alternatives> </jats:inline-formula> to metric measure spaces through limits of averaging integrals. The AMV Laplacian is however not a symmetric operator in general. Therefore, we consider a symmetric version of the AMV Laplacian, and focus lies on when the symmetric and non-symmetric AMV Laplacians coincide. Besides Riemannian and 3D contact sub-Riemannian manifolds, we show that they are identical on a large class of metric measure spaces, including locally Ahlfors regular spaces with suitably vanishing distortion. In addition, we study the context of weighted domains of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$mathbb {R}^n$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S030821052300118X_inline2.png\" /> </jats:alternatives> </jats:inline-formula>—where the two operators typically differ—and provide explicit formulae for these operators, including points where the weight vanishes.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":"327 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138517016","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study spaces of continuous functions and sections with domain a paracompact Hausdorff k-space $X$ and range a nilpotent CW complex $Y$, with emphasis on localization at a set of primes. For $mathop {rm map}nolimits _phi (X,,Y)$, the space of maps with prescribed restriction $phi$ on a suitable subspace $Asubset X$, we construct a natural spectral sequence of groups that converges to $pi _*(mathop {rm map}nolimits _phi (X,,Y))$ and allows for detection of localization on the level of $E^2$. Our applications extend and unify the previously known results.
{"title":"Spaces of functions and sections with paracompact domain","authors":"Jaka Smrekar","doi":"10.1017/prm.2023.117","DOIUrl":"https://doi.org/10.1017/prm.2023.117","url":null,"abstract":"We study spaces of continuous functions and sections with domain a paracompact Hausdorff <jats:italic>k</jats:italic>-space <jats:inline-formula> <jats:alternatives> <jats:tex-math>$X$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210523001178_inline1.png\" /> </jats:alternatives> </jats:inline-formula> and range a nilpotent CW complex <jats:inline-formula> <jats:alternatives> <jats:tex-math>$Y$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210523001178_inline2.png\" /> </jats:alternatives> </jats:inline-formula>, with emphasis on localization at a set of primes. For <jats:inline-formula> <jats:alternatives> <jats:tex-math>$mathop {rm map}nolimits _phi (X,,Y)$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210523001178_inline3.png\" /> </jats:alternatives> </jats:inline-formula>, the space of maps with prescribed restriction <jats:inline-formula> <jats:alternatives> <jats:tex-math>$phi$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210523001178_inline4.png\" /> </jats:alternatives> </jats:inline-formula> on a suitable subspace <jats:inline-formula> <jats:alternatives> <jats:tex-math>$Asubset X$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210523001178_inline5.png\" /> </jats:alternatives> </jats:inline-formula>, we construct a natural spectral sequence of groups that converges to <jats:inline-formula> <jats:alternatives> <jats:tex-math>$pi _*(mathop {rm map}nolimits _phi (X,,Y))$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210523001178_inline6.png\" /> </jats:alternatives> </jats:inline-formula> and allows for detection of localization on the level of <jats:inline-formula> <jats:alternatives> <jats:tex-math>$E^2$</jats:tex-math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0308210523001178_inline7.png\" /> </jats:alternatives> </jats:inline-formula>. Our applications extend and unify the previously known results.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":" 24","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the discrete Safronov-Dubovskiĭ aggregation equation associated with the physical condition, where particle injection and extraction take place in the dynamical system. In application, this model is used to describe the aggregation of particle-monomers in combination with sedimentation of particle-clusters. More precisely, we prove well-posedness of the considered model for a large class of aggregation kernel with source and efflux coefficients. Furthermore, over a long time period, we prove that the dynamical model attains a unique equilibrium solution with an exponential rate under a suitable condition on the forcing coefficient.
{"title":"Trend to equilibrium solution for the discrete Safronov–Dubovskiĭ aggregation equation with forcing","authors":"Arijit Das, Jitraj Saha","doi":"10.1017/prm.2023.116","DOIUrl":"https://doi.org/10.1017/prm.2023.116","url":null,"abstract":"We consider the discrete Safronov-Dubovskiĭ aggregation equation associated with the physical condition, where particle injection and extraction take place in the dynamical system. In application, this model is used to describe the aggregation of particle-monomers in combination with sedimentation of particle-clusters. More precisely, we prove well-posedness of the considered model for a large class of aggregation kernel with source and efflux coefficients. Furthermore, over a long time period, we prove that the dynamical model attains a unique equilibrium solution with an exponential rate under a suitable condition on the forcing coefficient.","PeriodicalId":54560,"journal":{"name":"Proceedings of the Royal Society of Edinburgh Section A-Mathematics","volume":" 25","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494471","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
-algebra $A$
has a nowhere scattered multiplier algebra $mathcal {M}(A)$
, that is, we study when $mathcal {M}(A)$
has no nonzero, elementary ideal-quotients. In particular, we prove that a $sigma$
-unital $mathrm {C}^*$
-algebra $A$
of
-comparison,
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