In this paper, we analyze L‐space surgeries on two component L‐space links. We show that if one surgery coefficient is negative for the L‐space surgery, then the corresponding link component is an unknot. If the link admits a very negative (that is, d1,d2≪0 ) L‐space surgery, it is either the unlink or the Hopf link. We also give a way to characterize the torus link T(2,2l) by observing an L‐space surgery Sd1,d23(L) with some d1d2<0 on a 2‐component L‐space link with unknotted components. For some 2‐component L‐space links, we give explicit descriptions of the L‐space surgery sets.
{"title":"L‐space surgeries on 2‐component L‐space links","authors":"Beibei Liu","doi":"10.1112/tlm3.12027","DOIUrl":"https://doi.org/10.1112/tlm3.12027","url":null,"abstract":"In this paper, we analyze L‐space surgeries on two component L‐space links. We show that if one surgery coefficient is negative for the L‐space surgery, then the corresponding link component is an unknot. If the link admits a very negative (that is, d1,d2≪0 ) L‐space surgery, it is either the unlink or the Hopf link. We also give a way to characterize the torus link T(2,2l) by observing an L‐space surgery Sd1,d23(L) with some d1d2<0 on a 2‐component L‐space link with unknotted components. For some 2‐component L‐space links, we give explicit descriptions of the L‐space surgery sets.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41714222","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 enumerate all orders in definite quaternion algebras over number fields with the Hermite property; this includes all orders with the cancellation property for locally free modules.
{"title":"Definite orders with locally free cancellation","authors":"Daniel Smertnig, J. Voight","doi":"10.1112/tlm3.12019","DOIUrl":"https://doi.org/10.1112/tlm3.12019","url":null,"abstract":"We enumerate all orders in definite quaternion algebras over number fields with the Hermite property; this includes all orders with the cancellation property for locally free modules.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2019-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12019","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42781259","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 extend some equilibrium‐type results first conjectured by Ambrus, Ball and Erdélyi, and then proved recenly by Hardin, Kendall and Saff. We work on the torus T≃[0,2π) , but the motivation comes from an analogous setup on the unit interval, investigated earlier by Fenton.
{"title":"A minimax problem for sums of translates on the torus","authors":"B. Farkas, B. Nagy, S. Révész","doi":"10.1112/tlm3.12010","DOIUrl":"https://doi.org/10.1112/tlm3.12010","url":null,"abstract":"We extend some equilibrium‐type results first conjectured by Ambrus, Ball and Erdélyi, and then proved recenly by Hardin, Kendall and Saff. We work on the torus T≃[0,2π) , but the motivation comes from an analogous setup on the unit interval, investigated earlier by Fenton.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48761599","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/tlm3.12008","DOIUrl":"https://doi.org/10.1112/tlm3.12008","url":null,"abstract":"","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43487237","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}
As a geometrical understanding of the maximal gravitational potential in computational and mathematical physics, this paper investigates a mixed volume induced by the so‐called anisotropic Riesz‐potential and establishes a reverse Minkowski‐type inequality. It turns out that such a mixed volume is equal to the anisotropic Riesz‐capacity and has connections with the anisotropic sup‐Riesz‐potential space. Two restrictions on the Lorentz spaces in terms of the anisotropic Riesz‐capacity are also characterized. Besides, we also prove a Minkowski‐type inequality and a log‐Minkowski‐type inequality as well as its reverse form.
{"title":"A mixed volume from the anisotropic Riesz‐potential","authors":"S. Hou, J. Xiao, Deping Ye","doi":"10.1112/tlm3.12012","DOIUrl":"https://doi.org/10.1112/tlm3.12012","url":null,"abstract":"As a geometrical understanding of the maximal gravitational potential in computational and mathematical physics, this paper investigates a mixed volume induced by the so‐called anisotropic Riesz‐potential and establishes a reverse Minkowski‐type inequality. It turns out that such a mixed volume is equal to the anisotropic Riesz‐capacity and has connections with the anisotropic sup‐Riesz‐potential space. Two restrictions on the Lorentz spaces in terms of the anisotropic Riesz‐capacity are also characterized. Besides, we also prove a Minkowski‐type inequality and a log‐Minkowski‐type inequality as well as its reverse form.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46791233","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}
The construction introduced by Gross, Hacking and Keel in (Several Complex Variables (Springer, New York, NY, 1976))allows one to construct a formal mirror family to a pair (S,D) where S is a smooth rational projective surface and D a certain type of Weil divisor supporting an ample or anti‐ample class. In that paper, they proved two convergence results when the intersection matrix of D is not negative semi‐definite and when the matrix is negative definite. In the original version of that paper, they claimed that if the intersection matrix were negative semi‐definite, then family extends over an analytic neighbourhood of the origin but gave an incorrect proof. In this paper, we correct this error. We reduce the construction of the mirror to such a surface to calculating certain log Gromov–Witten invariants. We then relate these invariants to the invariants of a new space where we can find explicit formulae for the invariants. From this we deduce analytic convergence of the mirror family, at least when the original surface has an I4 fibre.
{"title":"Convergence of the mirror to a rational elliptic surface","authors":"L. J. Barrott","doi":"10.1112/tlm3.12028","DOIUrl":"https://doi.org/10.1112/tlm3.12028","url":null,"abstract":"The construction introduced by Gross, Hacking and Keel in (Several Complex Variables (Springer, New York, NY, 1976))allows one to construct a formal mirror family to a pair (S,D) where S is a smooth rational projective surface and D a certain type of Weil divisor supporting an ample or anti‐ample class. In that paper, they proved two convergence results when the intersection matrix of D is not negative semi‐definite and when the matrix is negative definite. In the original version of that paper, they claimed that if the intersection matrix were negative semi‐definite, then family extends over an analytic neighbourhood of the origin but gave an incorrect proof. In this paper, we correct this error. We reduce the construction of the mirror to such a surface to calculating certain log Gromov–Witten invariants. We then relate these invariants to the invariants of a new space where we can find explicit formulae for the invariants. From this we deduce analytic convergence of the mirror family, at least when the original surface has an I4 fibre.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48812344","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}
The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic functions in several non‐commuting variables. In this paper we introduce the class of nc‐manifolds, the mathematical objects that at each point possess a neighborhood that has the structure of an nc‐domain in the d ‐dimensional nc‐universe Md . We illustrate the use of such manifolds in free analysis through the construction of the non‐commutative Riemann surface for the matricial square root function. A second illustration is the construction of a non‐commutative analog of the elementary symmetric functions in two variables. For any symmetric domain in M2 we construct a two‐dimensional non‐commutative manifold such that the symmetric holomorphic functions on the domain are in bijective correspondence with the holomorphic functions on the manifold. We also derive a version of the classical Newton–Girard formulae for power sums of two non‐commuting variables.
{"title":"Non‐commutative manifolds, the free square root and symmetric functions in two non‐commuting variables","authors":"J. Agler, John E. McCarthy, N. Young","doi":"10.1112/tlm3.12015","DOIUrl":"https://doi.org/10.1112/tlm3.12015","url":null,"abstract":"The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic functions in several non‐commuting variables. In this paper we introduce the class of nc‐manifolds, the mathematical objects that at each point possess a neighborhood that has the structure of an nc‐domain in the d ‐dimensional nc‐universe Md . We illustrate the use of such manifolds in free analysis through the construction of the non‐commutative Riemann surface for the matricial square root function. A second illustration is the construction of a non‐commutative analog of the elementary symmetric functions in two variables. For any symmetric domain in M2 we construct a two‐dimensional non‐commutative manifold such that the symmetric holomorphic functions on the domain are in bijective correspondence with the holomorphic functions on the manifold. We also derive a version of the classical Newton–Girard formulae for power sums of two non‐commuting variables.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12015","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49502393","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 study topological mixing properties and the maximal equicontinuous factor of rank‐one subshifts as topological dynamical systems. We show that the maximal equicontinuous factor of a rank‐one subshift is finite. We also determine all the finite factors of a rank‐one shift with a condition involving the cutting and spacer parameters. For rank‐one subshifts with bounded spacer parameter we completely characterize weak mixing and mixing. For rank‐one subshifts with unbounded spacer parameter we prove some sufficient conditions for weak mixing and mixing. We also construct some examples showing that the characterizations for the bounded spacer parameter case do not generalize to the unbounded spacer parameter case.
{"title":"Topological mixing properties of rank‐one subshifts","authors":"Su Gao, Caleb Ziegler","doi":"10.1112/tlm3.12016","DOIUrl":"https://doi.org/10.1112/tlm3.12016","url":null,"abstract":"We study topological mixing properties and the maximal equicontinuous factor of rank‐one subshifts as topological dynamical systems. We show that the maximal equicontinuous factor of a rank‐one subshift is finite. We also determine all the finite factors of a rank‐one shift with a condition involving the cutting and spacer parameters. For rank‐one subshifts with bounded spacer parameter we completely characterize weak mixing and mixing. For rank‐one subshifts with unbounded spacer parameter we prove some sufficient conditions for weak mixing and mixing. We also construct some examples showing that the characterizations for the bounded spacer parameter case do not generalize to the unbounded spacer parameter case.","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12016","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63412159","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}
Recently, there has been much interest in studying the torsion subgroups of elliptic curves base‐extended to infinite extensions of Q . In this paper, given a finite group G , we study what happens with the torsion of an elliptic curve E over Q when changing base to the compositum of all number fields with Galois group G . We do this by studying a group theoretic condition called generalized G ‐type, which is a necessary condition for a number field with Galois group H to be contained in that compositum. In general, group theory allows one to reduce the original problem to the question of finding rational points on finitely many modular curves. To illustrate this method, we completely determine which torsion structures occur for elliptic curves defined over Q and base‐changed to the compositum of all fields whose Galois group is A4 .
{"title":"Groups of generalized G ‐type and applications to torsion subgroups of rational elliptic curves over infinite extensions of Q","authors":"Harris B. Daniels, M. Derickx, Jeffrey Hatley","doi":"10.1112/tlm3.12018","DOIUrl":"https://doi.org/10.1112/tlm3.12018","url":null,"abstract":"Recently, there has been much interest in studying the torsion subgroups of elliptic curves base‐extended to infinite extensions of Q . In this paper, given a finite group G , we study what happens with the torsion of an elliptic curve E over Q when changing base to the compositum of all number fields with Galois group G . We do this by studying a group theoretic condition called generalized G ‐type, which is a necessary condition for a number field with Galois group H to be contained in that compositum. In general, group theory allows one to reduce the original problem to the question of finding rational points on finitely many modular curves. To illustrate this method, we completely determine which torsion structures occur for elliptic curves defined over Q and base‐changed to the compositum of all fields whose Galois group is A4 .","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2018-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12018","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41984171","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/tlm3.12000","DOIUrl":"https://doi.org/10.1112/tlm3.12000","url":null,"abstract":"","PeriodicalId":41208,"journal":{"name":"Transactions of the London Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2017-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/tlm3.12000","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42361631","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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