{"title":"On the existence of uniformly bounded self-adjoint bases in GNS spaces","authors":"Debabrata De, Kunal Mukherjee","doi":"10.4171/dm/941","DOIUrl":"https://doi.org/10.4171/dm/941","url":null,"abstract":"","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"26 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139212842","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}
{"title":"Corrigendum to “The variety of polar simplices”","authors":"Kristian Ranestad, F. Schreyer","doi":"10.4171/dm/943","DOIUrl":"https://doi.org/10.4171/dm/943","url":null,"abstract":"","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"57 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139212389","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}
{"title":"The locus of curves with an odd subcanonical point","authors":"André Contiero, Aislan Fontes","doi":"10.4171/dm/934","DOIUrl":"https://doi.org/10.4171/dm/934","url":null,"abstract":"","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"121 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134956886","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}
In this paper, we provide quantitative versions of results on the asymptotic behavior of nonlinear semigroups generated by an accretive operator due to O. Nevanlinna and S. Reich as well as H.-K. Xu. These results themselves rely on a particular assumption on the underlying operator introduced by A. Pazy under the name of `convergence condition'. Based on logical techniques from `proof mining', a subdiscipline of mathematical logic, we derive various notions of a `convergence condition with modulus' which provide quantitative information on this condition in different ways. These techniques then also facilitate the extraction of quantitative information on the convergence results of Nevanlinna and Reich as well as Xu, in particular also in the form of rates of convergence which depend on these moduli for the convergence condition.
{"title":"On computational properties of Cauchy problems generated by accretive operators","authors":"Pedro Pinto, Nicholas Pischke","doi":"10.4171/dm/924","DOIUrl":"https://doi.org/10.4171/dm/924","url":null,"abstract":"In this paper, we provide quantitative versions of results on the asymptotic behavior of nonlinear semigroups generated by an accretive operator due to O. Nevanlinna and S. Reich as well as H.-K. Xu. These results themselves rely on a particular assumption on the underlying operator introduced by A. Pazy under the name of `convergence condition'. Based on logical techniques from `proof mining', a subdiscipline of mathematical logic, we derive various notions of a `convergence condition with modulus' which provide quantitative information on this condition in different ways. These techniques then also facilitate the extraction of quantitative information on the convergence results of Nevanlinna and Reich as well as Xu, in particular also in the form of rates of convergence which depend on these moduli for the convergence condition.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"1 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134957036","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}
Indranil Biswas, Krishna Hanumanthu, Snehajit Misra, Nabanita Ray
Let $X$ be a complex projective variety, and let $E_{ast}$ be a parabolic vector bundle on $X$. We introduce the notion of textit{parabolic Seshadri constants} of $E_{ast}$. It is shown that these constants are analogous to the classical Seshadri constants of vector bundles, in particular, they have parallel definitions and properties. We prove a Seshadri criterion for parabolic ampleness of $E_{ast}$ in terms of parabolic Seshadri constants. We also compute parabolic Seshadri constants for symmetric powers and tensor products of parabolic vector bundles.
{"title":"Seshadri constants of parabolic vector bundles","authors":"Indranil Biswas, Krishna Hanumanthu, Snehajit Misra, Nabanita Ray","doi":"10.4171/dm/917","DOIUrl":"https://doi.org/10.4171/dm/917","url":null,"abstract":"Let $X$ be a complex projective variety, and let $E_{ast}$ be a parabolic vector bundle on $X$. We introduce the notion of textit{parabolic Seshadri constants} of $E_{ast}$. It is shown that these constants are analogous to the classical Seshadri constants of vector bundles, in particular, they have parallel definitions and properties. We prove a Seshadri criterion for parabolic ampleness of $E_{ast}$ in terms of parabolic Seshadri constants. We also compute parabolic Seshadri constants for symmetric powers and tensor products of parabolic vector bundles.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"121 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134956884","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 a spinless, non-relativistic particle bound by an external potential and linearly coupled to a quantized radiation field. The energy $mathcal{E}(u,f)$ of product states of the form $uotimes Psi_f$, where $u$ is a normalized state for the particle and $Psi_f$ is a coherent state in Fock space for the field, gives the energy of a Klein-Gordon--Schr''odinger system. We minimize the functional $mathcal{E}(u,f)$ on its natural energy space. We prove the existence and uniqueness of a ground state under general conditions on the coupling function. In particular, neither an ultraviolet cutoff nor an infrared cutoff is imposed. Our results establish the convergence in the ultraviolet limit of both the ground state and ground state energy of the Klein-Gordon--Schr''odinger energy functional, and provide the second-order asymptotic expansion of the ground state energy at small coupling.
{"title":"Quasi-classical ground states. I. Linearly coupled Pauli–Fierz Hamiltonians","authors":"Sébastien Breteaux, Jérémy Faupin, Jimmy Payet","doi":"10.4171/dm/929","DOIUrl":"https://doi.org/10.4171/dm/929","url":null,"abstract":"We consider a spinless, non-relativistic particle bound by an external potential and linearly coupled to a quantized radiation field. The energy $mathcal{E}(u,f)$ of product states of the form $uotimes Psi_f$, where $u$ is a normalized state for the particle and $Psi_f$ is a coherent state in Fock space for the field, gives the energy of a Klein-Gordon--Schr''odinger system. We minimize the functional $mathcal{E}(u,f)$ on its natural energy space. We prove the existence and uniqueness of a ground state under general conditions on the coupling function. In particular, neither an ultraviolet cutoff nor an infrared cutoff is imposed. Our results establish the convergence in the ultraviolet limit of both the ground state and ground state energy of the Klein-Gordon--Schr''odinger energy functional, and provide the second-order asymptotic expansion of the ground state energy at small coupling.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"4 24","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136229710","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}
{"title":"Erratum to “Itô’s formula for noncommutative $C^2$ functions of free Itô processes”","authors":"Evangelos A. Nikitopoulos","doi":"10.4171/dm/932","DOIUrl":"https://doi.org/10.4171/dm/932","url":null,"abstract":"","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"107 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139277073","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}
{"title":"Quasi-isogeny groups of supersingular abelian surfaces via pro-étale fundamental groups","authors":"Thibaud van den Hove","doi":"10.4171/dm/930","DOIUrl":"https://doi.org/10.4171/dm/930","url":null,"abstract":"","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"119 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134957732","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}
Trialitarian triples are triples of central simple algebras of degree 8 with orthogonal involution that provide a convenient structure for the representation of trialitarian algebraic groups as automorphism groups. This paper explicitly describes the canonical"trialitarian'' isomorphisms between the spin groups of the algebras with involution involved in a trialitarian triple, using a rationally defined shift operator that cyclically permutes the algebras. The construction relies on compositions of quadratic spaces of dimension 8, which yield all the trialitarian triples of split algebras. No restriction on the characteristic of the base field is needed.
{"title":"Trialitarian triples","authors":"Demba Barry, Jean-Pierre Tignol","doi":"10.4171/dm/926","DOIUrl":"https://doi.org/10.4171/dm/926","url":null,"abstract":"Trialitarian triples are triples of central simple algebras of degree 8 with orthogonal involution that provide a convenient structure for the representation of trialitarian algebraic groups as automorphism groups. This paper explicitly describes the canonical\"trialitarian'' isomorphisms between the spin groups of the algebras with involution involved in a trialitarian triple, using a rationally defined shift operator that cyclically permutes the algebras. The construction relies on compositions of quadratic spaces of dimension 8, which yield all the trialitarian triples of split algebras. No restriction on the characteristic of the base field is needed.","PeriodicalId":50567,"journal":{"name":"Documenta Mathematica","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136113210","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}
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