In this paper, we study a class of fractional Schrödinger equations involving logarithmic and critical non‐linearities on an unbounded domain, and show that such an equation with positive or sign‐changing weight potentials admits at least one positive ground state solution and the associated energy is positive (or negative). By applying the Nehari manifold method and Ljusternik–Schnirelmann category, we investigate how the weight potential affects the multiplicity of positive solutions, and obtain the relationship between the number of positive solutions and the category of some sets related to the weight potential.
{"title":"Positive solutions for the fractional Schrödinger equations with logarithmic and critical non‐linearities","authors":"H. Fan, Zhaosheng Feng, Xingjie Yan","doi":"10.1112/tlm3.12034","DOIUrl":"https://doi.org/10.1112/tlm3.12034","url":null,"abstract":"In this paper, we study a class of fractional Schrödinger equations involving logarithmic and critical non‐linearities on an unbounded domain, and show that such an equation with positive or sign‐changing weight potentials admits at least one positive ground state solution and the associated energy is positive (or negative). By applying the Nehari manifold method and Ljusternik–Schnirelmann category, we investigate how the weight potential affects the multiplicity of positive solutions, and obtain the relationship between the number of positive solutions and the category of some sets related to the weight potential.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":"8 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43083695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
C. Berkesch, Patricia Klein, Michael C. Loper, J. Yang
When studying a graded module M over the Cox ring of a smooth projective toric variety X , there are two standard types of resolutions commonly used to glean information: free resolutions of M and vector bundle resolutions of its sheafification. Each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions can resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman and Smith introduced virtual resolutions, which capture desirable geometric information and are also amenable to algebraic and combinatorial study. The theory of virtual resolutions includes a notion of a virtually Cohen–Macaulay property, though tools for assessing which modules are virtually Cohen–Macaulay have only recently started to be developed. In this article, we continue this research program in two related ways. The first is that, when X is a product of projective spaces, we produce a large new class of virtually Cohen–Macaulay Stanley–Reisner rings, which we show to be virtually Cohen–Macaulay via explicit constructions of appropriate virtual resolutions reflecting the underlying combinatorial structure. The second is that, for an arbitrary smooth projective toric variety X , we develop homological tools for assessing the virtual Cohen–Macaulay property. Some of these tools give exclusionary criteria, and others are constructive methods for producing suitably short virtual resolutions. We also use these tools to establish relationships among the arithmetically, geometrically and virtually Cohen–Macaulay properties.
{"title":"Homological and combinatorial aspects of virtually Cohen–Macaulay sheaves","authors":"C. Berkesch, Patricia Klein, Michael C. Loper, J. Yang","doi":"10.1112/tlm3.12036","DOIUrl":"https://doi.org/10.1112/tlm3.12036","url":null,"abstract":"When studying a graded module M over the Cox ring of a smooth projective toric variety X , there are two standard types of resolutions commonly used to glean information: free resolutions of M and vector bundle resolutions of its sheafification. Each approach comes with its own challenges. There is geometric information that free resolutions fail to encode, while vector bundle resolutions can resist study using algebraic and combinatorial techniques. Recently, Berkesch, Erman and Smith introduced virtual resolutions, which capture desirable geometric information and are also amenable to algebraic and combinatorial study. The theory of virtual resolutions includes a notion of a virtually Cohen–Macaulay property, though tools for assessing which modules are virtually Cohen–Macaulay have only recently started to be developed. In this article, we continue this research program in two related ways. The first is that, when X is a product of projective spaces, we produce a large new class of virtually Cohen–Macaulay Stanley–Reisner rings, which we show to be virtually Cohen–Macaulay via explicit constructions of appropriate virtual resolutions reflecting the underlying combinatorial structure. The second is that, for an arbitrary smooth projective toric variety X , we develop homological tools for assessing the virtual Cohen–Macaulay property. Some of these tools give exclusionary criteria, and others are constructive methods for producing suitably short virtual resolutions. We also use these tools to establish relationships among the arithmetically, geometrically and virtually Cohen–Macaulay properties.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47649399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that Mirzakhani's curve counting theorem also holds if we replace surfaces by orbifolds.
我们证明了Mirzakhani的曲线计数定理也成立，如果我们用轨道代替曲面。
{"title":"Counting curves on orbifolds","authors":"V. Erlandsson, J. Souto","doi":"10.1112/tlm3.12043","DOIUrl":"https://doi.org/10.1112/tlm3.12043","url":null,"abstract":"We show that Mirzakhani's curve counting theorem also holds if we replace surfaces by orbifolds.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":"9 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43434550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Totally symmetric self‐complementary plane partitions (TSSCPPs) are boxed plane partitions with the maximum possible symmetry. We use the well‐known representation of TSSCPPs as a dimer model on a honeycomb graph enclosed in 1/12 of a hexagon with free boundary to express them as perfect matchings of a family of non‐bipartite planar graphs. Our main result is that the edges of the TSSCPPs form a Pfaffian point process, for which we give explicit formulas for the inverse Kasteleyn matrix. Preliminary analysis of these correlations are then used to give a precise conjecture for the limit shape of TSSCPPs in the scaling limit.
{"title":"Correlations in totally symmetric self‐complementary plane partitions","authors":"Arvind Ayyer, S. Chhita","doi":"10.1112/tlm3.12039","DOIUrl":"https://doi.org/10.1112/tlm3.12039","url":null,"abstract":"Totally symmetric self‐complementary plane partitions (TSSCPPs) are boxed plane partitions with the maximum possible symmetry. We use the well‐known representation of TSSCPPs as a dimer model on a honeycomb graph enclosed in 1/12 of a hexagon with free boundary to express them as perfect matchings of a family of non‐bipartite planar graphs. Our main result is that the edges of the TSSCPPs form a Pfaffian point process, for which we give explicit formulas for the inverse Kasteleyn matrix. Preliminary analysis of these correlations are then used to give a precise conjecture for the limit shape of TSSCPPs in the scaling limit.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":"8 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41372501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by the study of symmetries of C∗ ‐algebras, as well as by multivariate operator theory, we introduce the notion of an SU(2) ‐equivariant subproduct system of Hilbert spaces. We analyse the resulting Toeplitz and Cuntz–Pimsner algebras and provide results about their topological invariants through Kasparov's bivariant K ‐theory. In particular, starting from an irreducible representation of SU(2) , we show that the corresponding Toeplitz algebra is equivariantly KK ‐equivalent to the algebra of complex numbers. In this way, we obtain a six‐term exact sequence of K ‐groups containing a noncommutative analogue of the Euler class.
{"title":"Gysin sequences and SU(2) ‐symmetries of C∗ ‐algebras","authors":"F. Arici, Jens Kaad","doi":"10.1112/tlm3.12038","DOIUrl":"https://doi.org/10.1112/tlm3.12038","url":null,"abstract":"Motivated by the study of symmetries of C∗ ‐algebras, as well as by multivariate operator theory, we introduce the notion of an SU(2) ‐equivariant subproduct system of Hilbert spaces. We analyse the resulting Toeplitz and Cuntz–Pimsner algebras and provide results about their topological invariants through Kasparov's bivariant K ‐theory. In particular, starting from an irreducible representation of SU(2) , we show that the corresponding Toeplitz algebra is equivariantly KK ‐equivalent to the algebra of complex numbers. In this way, we obtain a six‐term exact sequence of K ‐groups containing a noncommutative analogue of the Euler class.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49241162","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Issue Information","authors":"","doi":"10.1112/tlm3.12017","DOIUrl":"https://doi.org/10.1112/tlm3.12017","url":null,"abstract":"","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":"7 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45945339","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define twisted Eisenstein series Es±(h,k;τ) for s∈C , and show how their associated period functions, initially defined on the upper half complex plane H , have analytic continuation to all of C′:=C∖R⩽0 . We also use this result, as well as properties of various zeta functions, to show that certain cotangent‐zeta sums behave like quantum modular forms of (complex) weight s .
{"title":"Twisted Eisenstein series, cotangent‐zeta sums, and quantum modular forms","authors":"A. Folsom","doi":"10.1112/tlm3.12022","DOIUrl":"https://doi.org/10.1112/tlm3.12022","url":null,"abstract":"We define twisted Eisenstein series Es±(h,k;τ) for s∈C , and show how their associated period functions, initially defined on the upper half complex plane H , have analytic continuation to all of C′:=C∖R⩽0 . We also use this result, as well as properties of various zeta functions, to show that certain cotangent‐zeta sums behave like quantum modular forms of (complex) weight s .","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12022","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47348304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let G$G$ be a finite group and recall that the Frattini subgroup Frat(G)${rm Frat}(G)$ is the intersection of all the maximal subgroups of G$G$ . In this paper, we investigate the intersection number of G$G$ , denoted α(G)$alpha (G)$ , which is the minimal number of maximal subgroups whose intersection coincides with Frat(G)${rm Frat}(G)$ . In earlier work, we studied α(G)$alpha (G)$ in the special case where G$G$ is simple and here we extend the analysis to almost simple groups. In particular, we prove that α(G)⩽4$alpha (G) leqslant 4$ for every almost simple group G$G$ , which is best possible. We also establish new results on the intersection number of arbitrary finite groups, obtaining upper bounds that are defined in terms of the chief factors of the group. Finally, for almost simple groups G$G$ we present best possible bounds on a related invariant β(G)$beta (G)$ , which we call the base number of G$G$ . In this setting, β(G)$beta (G)$ is the minimal base size of G$G$ as we range over all faithful primitive actions of the group and we prove that the bound β(G)⩽4$beta (G) leqslant 4$ is optimal. Along the way, we study bases for the primitive action of the symmetric group Sab$S_{ab}$ on the set of partitions of [1,ab]$[1,ab]$ into a$a$ parts of size b$b$ , determining the exact base size for a⩾b$a geqslant b$ . This extends earlier work of Benbenishty, Cohen and Niemeyer.
{"title":"Finite groups, minimal bases and the intersection number","authors":"Timothy C. Burness, Martino Garonzi, A. Lucchini","doi":"10.1112/tlm3.12040","DOIUrl":"https://doi.org/10.1112/tlm3.12040","url":null,"abstract":"Let G$G$ be a finite group and recall that the Frattini subgroup Frat(G)${rm Frat}(G)$ is the intersection of all the maximal subgroups of G$G$ . In this paper, we investigate the intersection number of G$G$ , denoted α(G)$alpha (G)$ , which is the minimal number of maximal subgroups whose intersection coincides with Frat(G)${rm Frat}(G)$ . In earlier work, we studied α(G)$alpha (G)$ in the special case where G$G$ is simple and here we extend the analysis to almost simple groups. In particular, we prove that α(G)⩽4$alpha (G) leqslant 4$ for every almost simple group G$G$ , which is best possible. We also establish new results on the intersection number of arbitrary finite groups, obtaining upper bounds that are defined in terms of the chief factors of the group. Finally, for almost simple groups G$G$ we present best possible bounds on a related invariant β(G)$beta (G)$ , which we call the base number of G$G$ . In this setting, β(G)$beta (G)$ is the minimal base size of G$G$ as we range over all faithful primitive actions of the group and we prove that the bound β(G)⩽4$beta (G) leqslant 4$ is optimal. Along the way, we study bases for the primitive action of the symmetric group Sab$S_{ab}$ on the set of partitions of [1,ab]$[1,ab]$ into a$a$ parts of size b$b$ , determining the exact base size for a⩾b$a geqslant b$ . This extends earlier work of Benbenishty, Cohen and Niemeyer.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45488388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We complete the discrete cluster categories of type A as defined by Igusa and Todorov, by embedding such a discrete cluster category inside a larger one, and then taking a certain Verdier quotient. The resulting category is a Hom‐finite Krull–Schmidt triangulated category containing the discrete cluster category as a full subcategory. The objects and Hom‐spaces in this new category can be described geometrically, even though the category is not 2‐Calabi–Yau and Ext‐spaces are not always symmetric. We describe all cluster‐tilting subcategories. Given such a subcategory, we define a cluster character that takes values in a ring with infinitely many indeterminates. Our cluster character is new in that it takes into account infinite‐dimensional subrepresentations of infinite‐dimensional ones. We show that it satisfies the multiplication formula and also the exchange formula, provided that the objects being exchanged satisfy some local Calabi–Yau conditions.
{"title":"Completions of discrete cluster categories of type A","authors":"Charles Paquette, Emine Yildirim","doi":"10.1112/tlm3.12025","DOIUrl":"https://doi.org/10.1112/tlm3.12025","url":null,"abstract":"We complete the discrete cluster categories of type A as defined by Igusa and Todorov, by embedding such a discrete cluster category inside a larger one, and then taking a certain Verdier quotient. The resulting category is a Hom‐finite Krull–Schmidt triangulated category containing the discrete cluster category as a full subcategory. The objects and Hom‐spaces in this new category can be described geometrically, even though the category is not 2‐Calabi–Yau and Ext‐spaces are not always symmetric. We describe all cluster‐tilting subcategories. Given such a subcategory, we define a cluster character that takes values in a ring with infinitely many indeterminates. Our cluster character is new in that it takes into account infinite‐dimensional subrepresentations of infinite‐dimensional ones. We show that it satisfies the multiplication formula and also the exchange formula, provided that the objects being exchanged satisfy some local Calabi–Yau conditions.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48963383","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give a new algorithm for the construction of the unique superoptimal analytic approximant of a given continuous matrix‐valued function on the unit circle, using exterior powers of operators in preference to spectral or Wiener–Masani factorizations.
{"title":"Exterior products of operators and superoptimal analytic approximation","authors":"Dimitrios Chiotis, Z. Lykova, Nicholas Young","doi":"10.1112/tlm3.12035","DOIUrl":"https://doi.org/10.1112/tlm3.12035","url":null,"abstract":"We give a new algorithm for the construction of the unique superoptimal analytic approximant of a given continuous matrix‐valued function on the unit circle, using exterior powers of operators in preference to spectral or Wiener–Masani factorizations.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2020-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46605754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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