We discuss an approach to the secant non-defectivity of the varieties parametrising th powers of forms of degree . It employs a Terracini-type argument along with certain degeneration arguments, some of which are based on toric geometry. This implies a result on the identifiability of the Waring decompositions of general forms of degree kd as a sum of th powers of degree forms, for which an upper bound on the Waring rank was proposed by Fröberg, Ottaviani and Shapiro.
{"title":"Waring identifiability for powers of forms via degenerations","authors":"Alex Casarotti, Elisa Postinghel","doi":"10.1112/plms.12579","DOIUrl":"https://doi.org/10.1112/plms.12579","url":null,"abstract":"We discuss an approach to the secant non-defectivity of the varieties parametrising <math altimg=\"urn:x-wiley:00246115:media:plms12579:plms12579-math-0001\" display=\"inline\" location=\"graphic/plms12579-math-0001.png\">\u0000<semantics>\u0000<mi>k</mi>\u0000$k$</annotation>\u0000</semantics></math>th powers of forms of degree <math altimg=\"urn:x-wiley:00246115:media:plms12579:plms12579-math-0002\" display=\"inline\" location=\"graphic/plms12579-math-0002.png\">\u0000<semantics>\u0000<mi>d</mi>\u0000$d$</annotation>\u0000</semantics></math>. It employs a Terracini-type argument along with certain degeneration arguments, some of which are based on toric geometry. This implies a result on the identifiability of the Waring decompositions of general forms of degree kd as a sum of <math altimg=\"urn:x-wiley:00246115:media:plms12579:plms12579-math-0003\" display=\"inline\" location=\"graphic/plms12579-math-0003.png\">\u0000<semantics>\u0000<mi>k</mi>\u0000$k$</annotation>\u0000</semantics></math>th powers of degree <math altimg=\"urn:x-wiley:00246115:media:plms12579:plms12579-math-0004\" display=\"inline\" location=\"graphic/plms12579-math-0004.png\">\u0000<semantics>\u0000<mi>d</mi>\u0000$d$</annotation>\u0000</semantics></math> forms, for which an upper bound on the Waring rank was proposed by Fröberg, Ottaviani and Shapiro.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"16 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138826588","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A Latin square is a matrix of symbols such that each symbol occurs exactly once in each row and column. A Latin square is row-Hamiltonian if the permutation induced by each pair of distinct rows of is a full cycle permutation. Row-Hamiltonian Latin squares are equivalent to perfect 1-factorisations of complete bipartite graphs. For the first time, we exhibit a family of Latin squares that are row-Hamiltonian and also achieve precisely one of the related properties of being column-Hamiltonian or symbol-Hamiltonian. This family allows us to construct non-trivial, anti-associative, isotopically -closed loop varieties, solving an open problem posed by Falconer in 1970.
{"title":"Row-Hamiltonian Latin squares and Falconer varieties","authors":"Jack Allsop, Ian M. Wanless","doi":"10.1112/plms.12575","DOIUrl":"https://doi.org/10.1112/plms.12575","url":null,"abstract":"A <i>Latin square</i> is a matrix of symbols such that each symbol occurs exactly once in each row and column. A Latin square <math altimg=\"urn:x-wiley:00246115:media:plms12575:plms12575-math-0001\" display=\"inline\" location=\"graphic/plms12575-math-0001.png\">\u0000<semantics>\u0000<mi>L</mi>\u0000$L$</annotation>\u0000</semantics></math> is <i>row-Hamiltonian</i> if the permutation induced by each pair of distinct rows of <math altimg=\"urn:x-wiley:00246115:media:plms12575:plms12575-math-0002\" display=\"inline\" location=\"graphic/plms12575-math-0002.png\">\u0000<semantics>\u0000<mi>L</mi>\u0000$L$</annotation>\u0000</semantics></math> is a full cycle permutation. Row-Hamiltonian Latin squares are equivalent to perfect 1-factorisations of complete bipartite graphs. For the first time, we exhibit a family of Latin squares that are row-Hamiltonian and also achieve precisely one of the related properties of being column-Hamiltonian or symbol-Hamiltonian. This family allows us to construct non-trivial, anti-associative, isotopically <math altimg=\"urn:x-wiley:00246115:media:plms12575:plms12575-math-0003\" display=\"inline\" location=\"graphic/plms12575-math-0003.png\">\u0000<semantics>\u0000<mi>L</mi>\u0000$L$</annotation>\u0000</semantics></math>-closed loop varieties, solving an open problem posed by Falconer in 1970.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"9 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138826705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The Erdős–Hooley Delta function is defined for as . We prove that for all . This improves on earlier work of Hooley, Hall–Tenenbaum, and La Bretèche–Tenenbaum.
Erdős-Hooley函数定义为。我们为所有人证明这一点。这改进了Hooley, Hall-Tenenbaum和La bret che - tenenbaum的早期工作。
{"title":"An upper bound on the mean value of the Erdős–Hooley Delta function","authors":"Dimitris Koukoulopoulos, Terence Tao","doi":"10.1112/plms.12572","DOIUrl":"https://doi.org/10.1112/plms.12572","url":null,"abstract":"Abstract The Erdős–Hooley Delta function is defined for as . We prove that for all . This improves on earlier work of Hooley, Hall–Tenenbaum, and La Bretèche–Tenenbaum.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" 22","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135242040","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract It is well known that the relative commutant of every separable nuclear ‐subalgebra of the Calkin algebra has a unital copy of Cuntz algebra . We prove that the Calkin algebra has a separable ‐subalgebra whose relative commutant has no simple, unital, and noncommutative ‐subalgebra. On the other hand, the corona of every stable, separable ‐algebra that tensorially absorbs the Jiang–Su algebra has the property that the relative commutant of every separable ‐subalgebra contains a unital copy of . Analogous result holds for other strongly self‐absorbing ‐algebras. As an application, the Calkin algebra is not isomorphic to the corona of the stabilization of the Cuntz algebra , any other Kirchberg algebra, or even the corona of the stabilization of any unital, ‐stable ‐algebra.
{"title":"The Calkin algebra, Kazhdan's property (T), and strongly self‐absorbing C∗$mathrm{C}^*$‐algebras","authors":"Ilijas Farah","doi":"10.1112/plms.12569","DOIUrl":"https://doi.org/10.1112/plms.12569","url":null,"abstract":"Abstract It is well known that the relative commutant of every separable nuclear ‐subalgebra of the Calkin algebra has a unital copy of Cuntz algebra . We prove that the Calkin algebra has a separable ‐subalgebra whose relative commutant has no simple, unital, and noncommutative ‐subalgebra. On the other hand, the corona of every stable, separable ‐algebra that tensorially absorbs the Jiang–Su algebra has the property that the relative commutant of every separable ‐subalgebra contains a unital copy of . Analogous result holds for other strongly self‐absorbing ‐algebras. As an application, the Calkin algebra is not isomorphic to the corona of the stabilization of the Cuntz algebra , any other Kirchberg algebra, or even the corona of the stabilization of any unital, ‐stable ‐algebra.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"24 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135774186","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Carles Broto, Jesper Møller, Bob Oliver, Albert Ruiz
Abstract A saturated fusion system over a finite ‐group is a category whose objects are the subgroups of and whose morphisms are injective homomorphisms between the subgroups satisfying certain axioms. A fusion system over is realized by a finite group if is a Sylow ‐subgroup of and morphisms in the category are those induced by conjugation in . One recurrent question in this subject is to find criteria as to whether a given saturated fusion system is realizable or not. One main result in this paper is that a saturated fusion system is realizable if all of its components (in the sense of Aschbacher) are realizable. Another result is that all realizable fusion systems are tame: a finer condition on realizable fusion systems that involves describing automorphisms of a fusion system in terms of those of some group that realizes it. Stated in this way, these results depend on the classification of finite simple groups, but we also give more precise formulations whose proof is independent of the classification.
{"title":"Realizability and tameness of fusion systems","authors":"Carles Broto, Jesper Møller, Bob Oliver, Albert Ruiz","doi":"10.1112/plms.12571","DOIUrl":"https://doi.org/10.1112/plms.12571","url":null,"abstract":"Abstract A saturated fusion system over a finite ‐group is a category whose objects are the subgroups of and whose morphisms are injective homomorphisms between the subgroups satisfying certain axioms. A fusion system over is realized by a finite group if is a Sylow ‐subgroup of and morphisms in the category are those induced by conjugation in . One recurrent question in this subject is to find criteria as to whether a given saturated fusion system is realizable or not. One main result in this paper is that a saturated fusion system is realizable if all of its components (in the sense of Aschbacher) are realizable. Another result is that all realizable fusion systems are tame: a finer condition on realizable fusion systems that involves describing automorphisms of a fusion system in terms of those of some group that realizes it. Stated in this way, these results depend on the classification of finite simple groups, but we also give more precise formulations whose proof is independent of the classification.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"18 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135874751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let be an odd integer. We construct an anticyclotomic Euler system for certain cuspidal automorphic representations of unitary groups with signature .
设为奇数。我们构造了具有签名的酉群的某些倒自同构表示的反胞群欧拉系统。
{"title":"Anticyclotomic Euler systems for unitary groups","authors":"Andrew Graham, Syed Waqar Ali Shah","doi":"10.1112/plms.12566","DOIUrl":"https://doi.org/10.1112/plms.12566","url":null,"abstract":"Abstract Let be an odd integer. We construct an anticyclotomic Euler system for certain cuspidal automorphic representations of unitary groups with signature .","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"55 13","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136017524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let be an Anosov diffeomorphism whose linearization is irreducible. Assume that is also absolutely partially hyperbolic where a weak stable subbundle is considered as the center subbundle. We show that if the strong stable subbundle and the unstable subbundle are jointly integrable, then is dynamically coherent and all foliations match corresponding linear foliation under the conjugacy to the linearization . Moreover, admits the finest dominated splitting in the weak stable subbundle with dimensions matching those for , and it has spectral rigidity along all these subbundles. In dimension 4, we also obtain a similar result by grouping the weak stable and unstable subbundles together as a center subbundle and assuming joint integrability of the strong stable and unstable subbundles. As an application, we show that for every symplectic diffeomorphism that is ‐close to an irreducible nonconformal automorphism , the extremal subbundles of are jointly integrable if and only if is smoothly conjugate to .
{"title":"Joint integrability and spectral rigidity for Anosov diffeomorphisms","authors":"Andrey Gogolev, Yi Shi","doi":"10.1112/plms.12568","DOIUrl":"https://doi.org/10.1112/plms.12568","url":null,"abstract":"Abstract Let be an Anosov diffeomorphism whose linearization is irreducible. Assume that is also absolutely partially hyperbolic where a weak stable subbundle is considered as the center subbundle. We show that if the strong stable subbundle and the unstable subbundle are jointly integrable, then is dynamically coherent and all foliations match corresponding linear foliation under the conjugacy to the linearization . Moreover, admits the finest dominated splitting in the weak stable subbundle with dimensions matching those for , and it has spectral rigidity along all these subbundles. In dimension 4, we also obtain a similar result by grouping the weak stable and unstable subbundles together as a center subbundle and assuming joint integrability of the strong stable and unstable subbundles. As an application, we show that for every symplectic diffeomorphism that is ‐close to an irreducible nonconformal automorphism , the extremal subbundles of are jointly integrable if and only if is smoothly conjugate to .","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"34 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135112661","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract It is a question of Hochman whether any two one‐dimensional topologically mixing subshifts of finite type (SFTs) with the same entropy are topologically conjugate when their periodic points are removed. We give a negative answer, and, in fact, we prove the stronger result that there is a canonical correspondence between topological conjugacies of transitive SFTs and topological conjugacies between the systems obtained by removing the periodic points.
{"title":"Conjugacy of transitive SFTs minus periodic points","authors":"Ville Salo","doi":"10.1112/plms.12567","DOIUrl":"https://doi.org/10.1112/plms.12567","url":null,"abstract":"Abstract It is a question of Hochman whether any two one‐dimensional topologically mixing subshifts of finite type (SFTs) with the same entropy are topologically conjugate when their periodic points are removed. We give a negative answer, and, in fact, we prove the stronger result that there is a canonical correspondence between topological conjugacies of transitive SFTs and topological conjugacies between the systems obtained by removing the periodic points.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"53 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135217343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider measurable functions on that tile simultaneously by two arithmetic progressions and at respective tiling levels and . We are interested in two main questions: what are the possible values of the tiling levels , and what is the least possible measure of the support of ? We obtain sharp results which show that the answers depend on arithmetic properties of and , and in particular, on whether the numbers are rationally independent or not.
{"title":"Functions tiling simultaneously with two arithmetic progressions","authors":"Mark Mordechai Etkind, Nir Lev","doi":"10.1112/plms.12570","DOIUrl":"https://doi.org/10.1112/plms.12570","url":null,"abstract":"Abstract We consider measurable functions on that tile simultaneously by two arithmetic progressions and at respective tiling levels and . We are interested in two main questions: what are the possible values of the tiling levels , and what is the least possible measure of the support of ? We obtain sharp results which show that the answers depend on arithmetic properties of and , and in particular, on whether the numbers are rationally independent or not.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135267617","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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