Pub Date : 2025-01-10DOI: 10.1016/j.geomphys.2025.105422
Guangwen Zhao
In the case where both the domain and target manifolds are almost Hermitian, we introduce the concept of Hermitian pluriharmonic maps. We prove that any holomorphic or anti-holomorphic map between almost Hermitian manifolds is Hermitian pluriharmonic. We also establish some monotonicity formulae for the partial energies of Hermitian pluriharmonic maps into Kähler manifolds. As an application, under appropriate assumptions on the growth of the partial energies, some holomorphicity results are proven. When the domain manifold degenerates into Kähler and Hermitian, our results partially improve upon those of Dong (2013) [5] and Yang et al. (2013) [22], respectively.
{"title":"Hermitian pluriharmonic maps between almost Hermitian manifolds","authors":"Guangwen Zhao","doi":"10.1016/j.geomphys.2025.105422","DOIUrl":"10.1016/j.geomphys.2025.105422","url":null,"abstract":"<div><div>In the case where both the domain and target manifolds are almost Hermitian, we introduce the concept of Hermitian pluriharmonic maps. We prove that any holomorphic or anti-holomorphic map between almost Hermitian manifolds is Hermitian pluriharmonic. We also establish some monotonicity formulae for the partial energies of Hermitian pluriharmonic maps into Kähler manifolds. As an application, under appropriate assumptions on the growth of the partial energies, some holomorphicity results are proven. When the domain manifold degenerates into Kähler and Hermitian, our results partially improve upon those of Dong (2013) <span><span>[5]</span></span> and Yang et al. (2013) <span><span>[22]</span></span>, respectively.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105422"},"PeriodicalIF":1.6,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096615","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}
Pub Date : 2025-01-10DOI: 10.1016/j.geomphys.2025.105419
Alfonso García-Parrado
We introduce a new family of operators in 4-dimensional pseudo-Riemannian manifolds with a non-vanishing Weyl scalar (non-degenerate spaces) that keep the conformal covariance of conformally covariant tensor concomitants. A particular case that arises naturally is the -connection that is a Weyl connection that keeps conformal invariance. Using the connection we give a new characterization of non-degenerate spaces that are conformal to an Einstein space.
{"title":"The C-connection and the 4-dimensional Einstein spaces","authors":"Alfonso García-Parrado","doi":"10.1016/j.geomphys.2025.105419","DOIUrl":"10.1016/j.geomphys.2025.105419","url":null,"abstract":"<div><div>We introduce a new family of operators in 4-dimensional pseudo-Riemannian manifolds with a non-vanishing Weyl scalar (non-degenerate spaces) that keep the conformal covariance of <em>conformally covariant tensor concomitants</em>. A particular case that arises naturally is the <span><math><mi>C</mi></math></span>-connection that is a Weyl connection that keeps <em>conformal invariance</em>. Using the <span><math><mi>C</mi><mo>−</mo></math></span>connection we give a new characterization of non-degenerate spaces that are conformal to an Einstein space.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"210 ","pages":"Article 105419"},"PeriodicalIF":1.6,"publicationDate":"2025-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143159633","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}
Pub Date : 2025-01-09DOI: 10.1016/j.geomphys.2025.105417
Gülistan Polat , Jae Won Lee , Bayram Şahin
In this paper, Casorati inequalities are obtained for Riemannian maps and Riemannian submersions defined on Sasakian manifolds, and geometric results are given for the equality cases. First, Casorati inequalities for a Riemannian map from a Sasakian space form to a Riemannian manifold are obtained, and the equality case holds from geometric properties. Afterwards, Casorati inequalities involving tensor fields T and A are obtained for a Riemannian submersion from a Sasakian space form to a Riemann manifold, and geometric interpretations are given. It is shown that the equality of the inequalities obtained for tensor field A is equivalent to the integrability of the horizontal distribution. In the last section, Casorati inequalities and geometric results of a Riemannian map from a Sasakian manifold to a Sasakian space form are given.
{"title":"Optimal inequalities involving Casorati curvatures along Riemannian maps and Riemannian submersions for Sasakian space forms","authors":"Gülistan Polat , Jae Won Lee , Bayram Şahin","doi":"10.1016/j.geomphys.2025.105417","DOIUrl":"10.1016/j.geomphys.2025.105417","url":null,"abstract":"<div><div>In this paper, Casorati inequalities are obtained for Riemannian maps and Riemannian submersions defined on Sasakian manifolds, and geometric results are given for the equality cases. First, Casorati inequalities for a Riemannian map from a Sasakian space form to a Riemannian manifold are obtained, and the equality case holds from geometric properties. Afterwards, Casorati inequalities involving tensor fields <em>T</em> and <em>A</em> are obtained for a Riemannian submersion from a Sasakian space form to a Riemann manifold, and geometric interpretations are given. It is shown that the equality of the inequalities obtained for tensor field <em>A</em> is equivalent to the integrability of the horizontal distribution. In the last section, Casorati inequalities and geometric results of a Riemannian map from a Sasakian manifold to a Sasakian space form are given.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"210 ","pages":"Article 105417"},"PeriodicalIF":1.6,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143159637","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}
Pub Date : 2025-01-09DOI: 10.1016/j.geomphys.2025.105418
José F. Cariñena , Partha Guha
A locally conformal symplectic structure (for short l.c.s.) on a smooth manifold M is a generalisation of a symplectic structure. In this paper at first the theory of locally conformal symplectic structures is reviewed, and a description of the Lichnerowicz-Witten (LW) deformed differential operator is given. Using the exterior algebra of the LW differential operator, Hamiltonian vector fields associated to such l.c.s. structures are introduced. Several useful identities of the deformed exterior calculus are derived. The theory of symmetries of such locally conformal symplectic structures is developed. We show examples of the applications of our formalism, in particular, we present nonholonomic oscillator equation which admits a locally conformal symplectic structure. We study canonoid transformations of a locally Hamiltonian vector field on a locally conformal symplectic manifold. In particular, we present a generalized geometric theory of canonoid transformation in the l.c.s. structure setting.
{"title":"Lichnerowicz-Witten differential, symmetries and locally conformal symplectic structures","authors":"José F. Cariñena , Partha Guha","doi":"10.1016/j.geomphys.2025.105418","DOIUrl":"10.1016/j.geomphys.2025.105418","url":null,"abstract":"<div><div>A locally conformal symplectic structure (for short l.c.s.) on a smooth manifold <em>M</em> is a generalisation of a symplectic structure. In this paper at first the theory of locally conformal symplectic structures is reviewed, and a description of the Lichnerowicz-Witten (LW) deformed differential operator is given. Using the exterior algebra of the LW differential operator, Hamiltonian vector fields associated to such l.c.s. structures are introduced. Several useful identities of the deformed exterior calculus are derived. The theory of symmetries of such locally conformal symplectic structures is developed. We show examples of the applications of our formalism, in particular, we present nonholonomic oscillator equation which admits a locally conformal symplectic structure. We study canonoid transformations of a locally Hamiltonian vector field on a locally conformal symplectic manifold. In particular, we present a generalized geometric theory of canonoid transformation in the l.c.s. structure setting.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"210 ","pages":"Article 105418"},"PeriodicalIF":1.6,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143159634","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}
Pub Date : 2025-01-09DOI: 10.1016/j.geomphys.2025.105423
Alonso Perez-Lona
In this paper we discuss generalizations of discrete torsion to noninvertible symmetries in 2d QFTs. One point of this paper is to explain that there are two complementary generalizations. Both generalizations are counted by when one specializes to ordinary finite groups G. However, the counting is different for more general fusion categories. Furthermore, only one generalizes the picture of discrete torsion as differences in choices of gauge actions on B fields. Explaining this in detail, how one of the generalizations of discrete torsion to noninvertible cases encodes actions on B fields, is the other point of this paper. In particular, this generalizes old results in ordinary orbifolds that discrete torsion is a choice of group action on the B field. We also explain how this same generalization of discrete torsion gives rise to physically-sensible twists on gaugeable algebras and fiber functors.
{"title":"Discrete torsion in gauging non-invertible symmetries","authors":"Alonso Perez-Lona","doi":"10.1016/j.geomphys.2025.105423","DOIUrl":"10.1016/j.geomphys.2025.105423","url":null,"abstract":"<div><div>In this paper we discuss generalizations of discrete torsion to noninvertible symmetries in 2d QFTs. One point of this paper is to explain that there are two complementary generalizations. Both generalizations are counted by <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>G</mi><mo>,</mo><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo></math></span> when one specializes to ordinary finite groups <em>G</em>. However, the counting is different for more general fusion categories. Furthermore, only one generalizes the picture of discrete torsion as differences in choices of gauge actions on B fields. Explaining this in detail, how one of the generalizations of discrete torsion to noninvertible cases encodes actions on B fields, is the other point of this paper. In particular, this generalizes old results in ordinary orbifolds that discrete torsion is a choice of group action on the B field. We also explain how this same generalization of discrete torsion gives rise to physically-sensible twists on gaugeable algebras and fiber functors.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"210 ","pages":"Article 105423"},"PeriodicalIF":1.6,"publicationDate":"2025-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143159638","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}
Pub Date : 2025-01-06DOI: 10.1016/j.geomphys.2024.105416
Marc Mars, Gabriel Sánchez-Pérez
This is the first in a series of two papers where we analyze the transverse expansion of the metric on a general null hypersurface. In this paper we obtain general geometric identities relating the transverse derivatives of the ambient Ricci tensor and the transverse expansion of the metric at the null hypersurface. We also explore the case where the hypersurface exhibits a generalized symmetry generator, namely a privileged vector field in the ambient space which, at the hypersurface, is null and tangent (including the possibility of zeroes). This covers the Killing, homothetic, or conformal horizon cases, and, more generally, any situation where detailed information on the deformation tensor of the symmetry generator is available. Our approach is entirely covariant, independent on any field equations, and does not make any assumptions regarding the topology or dimension of the null hypersurface. As an application we prove that the full transverse expansion of the spacetime metric at a non-degenerate Killing horizon (also allowing for bifurcation surfaces) is uniquely determined in terms of abstract data on the horizon and the tower of derivatives of the ambient Ricci tensor at the horizon. In particular, the transverse expansion of the metric in Λ-vacuum spacetimes admitting a non-degenerate horizon is uniquely determined in terms of abstract data at the horizon.
{"title":"Transverse expansion of the metric at null hypersurfaces I. Uniqueness and application to Killing horizons","authors":"Marc Mars, Gabriel Sánchez-Pérez","doi":"10.1016/j.geomphys.2024.105416","DOIUrl":"10.1016/j.geomphys.2024.105416","url":null,"abstract":"<div><div>This is the first in a series of two papers where we analyze the transverse expansion of the metric on a general null hypersurface. In this paper we obtain general geometric identities relating the transverse derivatives of the ambient Ricci tensor and the transverse expansion of the metric at the null hypersurface. We also explore the case where the hypersurface exhibits a generalized symmetry generator, namely a privileged vector field in the ambient space which, at the hypersurface, is null and tangent (including the possibility of zeroes). This covers the Killing, homothetic, or conformal horizon cases, and, more generally, any situation where detailed information on the deformation tensor of the symmetry generator is available. Our approach is entirely covariant, independent on any field equations, and does not make any assumptions regarding the topology or dimension of the null hypersurface. As an application we prove that the full transverse expansion of the spacetime metric at a non-degenerate Killing horizon (also allowing for bifurcation surfaces) is uniquely determined in terms of abstract data on the horizon and the tower of derivatives of the ambient Ricci tensor at the horizon. In particular, the transverse expansion of the metric in Λ-vacuum spacetimes admitting a non-degenerate horizon is uniquely determined in terms of abstract data at the horizon.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105416"},"PeriodicalIF":1.6,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096614","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-31DOI: 10.1016/j.geomphys.2024.105415
Guo Chuan Thiang
The lowest Landau level Hilbert space, or the Bargmann–Fock space, admits a quantized trace for the commutator of its position coordinate operators. We exploit the Carey–Pincus theory of principal functions of trace class commutators to probe this integer quantization result further, and uncover a hidden rational structure in the higher-order commutator-traces. This shows how exact fractional quantization can occur whenever exact integral quantization does.
{"title":"Fractional index of Bargmann–Fock space and Landau levels","authors":"Guo Chuan Thiang","doi":"10.1016/j.geomphys.2024.105415","DOIUrl":"10.1016/j.geomphys.2024.105415","url":null,"abstract":"<div><div>The lowest Landau level Hilbert space, or the Bargmann–Fock space, admits a quantized trace for the commutator of its position coordinate operators. We exploit the Carey–Pincus theory of principal functions of trace class commutators to probe this integer quantization result further, and uncover a hidden rational structure in the higher-order commutator-traces. This shows how exact fractional quantization can occur whenever exact integral quantization does.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105415"},"PeriodicalIF":1.6,"publicationDate":"2024-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096612","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}
Pub Date : 2024-12-27DOI: 10.1016/j.geomphys.2024.105406
Iñaki Garay , Etera R. Livine
A geometric formula for the zeros of the partition function of the inhomogeneous 2d Ising model was recently proposed in terms of the angles of 2d triangulations embedded in the flat 3d space. Here we proceed to an analytical check of this formula on the cubic graph, dual to a double pyramid, and provide a thorough numerical check by generating random 2d planar triangulations. Our method is to generate Delaunay triangulations of the 2-sphere then performing random local rescalings. For every 2d triangulations, we compute the corresponding Ising couplings from the triangle angles and the dihedral angles, and check directly that the Ising partition function vanishes for these couplings (and grows in modulus in their neighborhood). In particular, we lift an ambiguity of the original formula on the sign of the dihedral angles and establish a convention in terms of convexity/concavity. Finally, we extend our numerical analysis to 2d toroidal triangulations and show that the geometric formula does not work and will need to be generalized, as originally expected, in order to accommodate for non-trivial topologies.
{"title":"Geometric formula for 2d Ising zeros: Examples & numerics","authors":"Iñaki Garay , Etera R. Livine","doi":"10.1016/j.geomphys.2024.105406","DOIUrl":"10.1016/j.geomphys.2024.105406","url":null,"abstract":"<div><div>A geometric formula for the zeros of the partition function of the inhomogeneous 2d Ising model was recently proposed in terms of the angles of 2d triangulations embedded in the flat 3d space. Here we proceed to an analytical check of this formula on the cubic graph, dual to a double pyramid, and provide a thorough numerical check by generating random 2d planar triangulations. Our method is to generate Delaunay triangulations of the 2-sphere then performing random local rescalings. For every 2d triangulations, we compute the corresponding Ising couplings from the triangle angles and the dihedral angles, and check directly that the Ising partition function vanishes for these couplings (and grows in modulus in their neighborhood). In particular, we lift an ambiguity of the original formula on the sign of the dihedral angles and establish a convention in terms of convexity/concavity. Finally, we extend our numerical analysis to 2d toroidal triangulations and show that the geometric formula does not work and will need to be generalized, as originally expected, in order to accommodate for non-trivial topologies.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105406"},"PeriodicalIF":1.6,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096611","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-27DOI: 10.1016/j.geomphys.2024.105414
Josef Mikeš , Sergey Stepanov , Irina Tsyganok
In present article, we consider a -orthogonal decomposition of the traceless part of the Ricci tensor of a closed Riemannian manifold and study its application to the geometry of compact Ricci almost solitons. In addition, we consider a -orthogonal expansion of the traceless part of the second fundamental form of a closed spacelike hypersurface in a Lorentzian manifold and study its application to the problem of constructing solutions of general relativistic vacuum constraint equations. In these two cases, we use the well-known Ahlfors Laplacian.
{"title":"New applications of the Ahlfors Laplacian: Ricci almost solitons and general relativistic vacuum constraint equations","authors":"Josef Mikeš , Sergey Stepanov , Irina Tsyganok","doi":"10.1016/j.geomphys.2024.105414","DOIUrl":"10.1016/j.geomphys.2024.105414","url":null,"abstract":"<div><div>In present article, we consider a <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-orthogonal decomposition of the traceless part of the Ricci tensor of a closed Riemannian manifold and study its application to the geometry of compact Ricci almost solitons. In addition, we consider a <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-orthogonal expansion of the traceless part of the second fundamental form of a closed spacelike hypersurface in a Lorentzian manifold and study its application to the problem of constructing solutions of general relativistic vacuum constraint equations. In these two cases, we use the well-known Ahlfors Laplacian.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105414"},"PeriodicalIF":1.6,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096617","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}
Pub Date : 2024-12-27DOI: 10.1016/j.geomphys.2024.105412
Charlotte Kirchhoff-Lukat
This article provides the first extension of Lagrangian Intersection Floer cohomology to Poisson structures which are almost everywhere symplectic, but degenerate on a lower-dimensional submanifold. The main result of the article is the definition of Lagrangian intersection Floer cohomology, referred to as log Floer cohomology, for orientable surfaces equipped with log symplectic structures. We show that this cohomology is invariant under suitable isotopies and that it is isomorphic to the log de Rham cohomology when computed for a single Lagrangian.
{"title":"Log Floer cohomology for oriented log symplectic surfaces","authors":"Charlotte Kirchhoff-Lukat","doi":"10.1016/j.geomphys.2024.105412","DOIUrl":"10.1016/j.geomphys.2024.105412","url":null,"abstract":"<div><div>This article provides the first extension of Lagrangian Intersection Floer cohomology to Poisson structures which are almost everywhere symplectic, but degenerate on a lower-dimensional submanifold. The main result of the article is the definition of Lagrangian intersection Floer cohomology, referred to as log Floer cohomology, for orientable surfaces equipped with log symplectic structures. We show that this cohomology is invariant under suitable isotopies and that it is isomorphic to the log de Rham cohomology when computed for a single Lagrangian.</div></div>","PeriodicalId":55602,"journal":{"name":"Journal of Geometry and Physics","volume":"209 ","pages":"Article 105412"},"PeriodicalIF":1.6,"publicationDate":"2024-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143096613","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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