1,747 results
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2. On Topological Properties of Some Coverings. An Addendum to a Paper of Lanteri and Struppa
- Author
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Jarosław A. Wiśniewski
- Subjects
Surjective function ,Ample line bundle ,Pure mathematics ,Morphism ,Betti number ,General Mathematics ,Embedding ,Projective space ,Projective test ,Space (mathematics) ,Mathematics - Abstract
Let π: X′ → X be a finite surjective morphism of complex projective manifolds which can be factored by an embedding of X′ into the total space of an ample line bundle 𝓛 over X. A theorem of Lazarsfeld asserts that Betti numbers of X and X′ are equal except, possibly, the middle ones. In the present paper it is proved that the middle numbers are actually non-equal if either 𝓛 is spanned and deg π ≥ dim X, or if X is either a hyperquadric or a projective space and π is not a double cover of an odd-dimensional projective space by a hyperquadric.
- Published
- 1992
3. On a Paper of Maurice Sion
- Author
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Mark Mahowald
- Subjects
Combinatorics ,Set (abstract data type) ,Class (set theory) ,Continuous function ,General Mathematics ,Bounded variation ,Open set ,Function (mathematics) ,Real line ,Measure (mathematics) ,Mathematics - Abstract
Let M0 be the set of measures μ on the real line such that open sets are μ*-immeasurable. While attempting to find out whether a set μ*-measurable for all μ in Mo is mapped into a similar set by a continuous function of bounded variation, Maurice Sion develops a theory for what he calls variational measure (4). As an application of the theory, he gets conditions on a function f and a set of measures M in order that f map a set, which is μ*-measurable for all μ ∈ M, into a set of the same kind. In particular he proves for his class M2 (def. 2.5), the following theorem (4, § 8.11).
- Published
- 1959
4. Note on a Paper by Robinson
- Author
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J. A. Todd
- Subjects
General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematical economics ,Mathematics - Abstract
In a recent paper Robinson has obtained an explicit formula for the expression of an invariant matrix of an invariant matrix as a direct sum of invariant matrices. The object of the present note is to show that this formula may be deduced from known properties of Schur functions, with the aid of a result which the author has proved elsewhere.
- Published
- 1950
5. Remark on my Paper 'Generators of Monothetic Groups'
- Author
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D. L. Armacost
- Subjects
Algebra ,General Mathematics ,Mathematics - Published
- 1973
6. Correction to the Paper*
- Author
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Casper Goffman and G. M. Petersen
- Subjects
Matrix (mathematics) ,Pure mathematics ,General Mathematics ,Arithmetic ,Mathematics - Published
- 1962
7. Unitary representations of type B rational Cherednik algebras and crystal combinatorics
- Author
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Emily Norton
- Subjects
Functor ,Unitarity ,General Mathematics ,Type (model theory) ,Unitary state ,Fock space ,Combinatorics ,Irreducible representation ,FOS: Mathematics ,Mathematics - Combinatorics ,Partition (number theory) ,Component (group theory) ,Combinatorics (math.CO) ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
We compare crystal combinatorics of the level 2 Fock space with the classification of unitary irreducible representations of type B rational Cherednik algebras to study how unitarity behaves under parabolic restriction. First, we show that any finite-dimensional unitary irreducible representation of such an algebra is labeled by a bipartition consisting of a rectangular partition in one component and the empty partition in the other component. This is a new proof of a result that can be deduced from theorems of Montarani and Etingof-Stoica. Second, we show that the crystal operators that remove boxes preserve the combinatorial conditions for unitarity, and that the parabolic restriction functors categorifying the crystals send irreducible unitary representations to unitary representations. Third, we find the supports of the unitary representations., This paper supersedes arXiv:1907.00919 and contains that paper as a subsection. 35 pages, some color figures
- Published
- 2021
8. Non-cocompact Group Actions and -Semistability at Infinity
- Author
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Ross Geoghegan, Michael L. Mihalik, and Craig R. Guilbault
- Subjects
Class (set theory) ,Pure mathematics ,Property (philosophy) ,Group (mathematics) ,General Mathematics ,media_common.quotation_subject ,010102 general mathematics ,Infinity ,01 natural sciences ,Group action ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics ,Counterexample ,media_common - Abstract
A finitely presented 1-ended group $G$ has semistable fundamental group at infinity if $G$ acts geometrically on a simply connected and locally compact ANR $Y$ having the property that any two proper rays in $Y$ are properly homotopic. This property of $Y$ captures a notion of connectivity at infinity stronger than “1-ended”, and is in fact a feature of $G$, being independent of choices. It is a fundamental property in the homotopical study of finitely presented groups. While many important classes of groups have been shown to have semistable fundamental group at infinity, the question of whether every $G$ has this property has been a recognized open question for nearly forty years. In this paper we attack the problem by considering a proper but non-cocompact action of a group $J$ on such an $Y$. This $J$ would typically be a subgroup of infinite index in the geometrically acting over-group $G$; for example $J$ might be infinite cyclic or some other subgroup whose semistability properties are known. We divide the semistability property of $G$ into a $J$-part and a “perpendicular to $J$” part, and we analyze how these two parts fit together. Among other things, this analysis leads to a proof (in a companion paper) that a class of groups previously considered to be likely counter examples do in fact have the semistability property.
- Published
- 2019
9. Corrigendum to: A Galois Correspondence for Reduced Crossed Products of Simple -algebras by Discrete Groups
- Author
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Roger R. Smith and Jan Cameron
- Subjects
Pure mathematics ,Crossed product ,Group (mathematics) ,Simple (abstract algebra) ,General Mathematics ,Unital ,Bimodule ,Mathematics - Abstract
This note corrects an error in our paper “A Galois correspondence for reduced crossed products of unital simple $\text{C}^{\ast }$-algebras by discrete groups”, http://dx.doi.org/10.4153/CJM-2018-014-6. The main results of the original paper are unchanged.
- Published
- 2019
10. Lorentz Estimates for Weak Solutions of Quasi-linear Parabolic Equations with Singular Divergence-free Drifts
- Author
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Tuoc Phan
- Subjects
General Mathematics ,Lorentz transformation ,010102 general mathematics ,Mathematical analysis ,Boundary (topology) ,Space (mathematics) ,01 natural sciences ,Parabolic partial differential equation ,010101 applied mathematics ,symbols.namesake ,Bounded function ,symbols ,Vector field ,Maximal function ,0101 mathematics ,Divergence (statistics) ,Mathematics - Abstract
This paper investigates regularity in Lorentz spaces for weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions that are measurable in ($x,t$)-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John–Nirenberg space. The results are even new when the drifts are identically zero, because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón–Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a “double-scaling parameter” technique and the maximal function free approach introduced by Acerbi and Mingione.
- Published
- 2019
11. Tannakian Categories With Semigroup Actions
- Author
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Michael Wibmer and Alexey Ovchinnikov
- Subjects
Class (set theory) ,Pure mathematics ,Semigroup ,General Mathematics ,010102 general mathematics ,Braid group ,Tannakian category ,Group Theory (math.GR) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,010101 applied mathematics ,Linear differential equation ,Mathematics::Category Theory ,FOS: Mathematics ,0101 mathematics ,Mathematics - Group Theory ,Finite set ,Differential (mathematics) ,Axiom ,Mathematics - Abstract
Ostrowski's theorem implies that $\log(x),\log(x+1),\ldots$ are algebraically independent over $\mathbb{C}(x)$. More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution $y$ and particular transformations of $y$, such as derivatives of $y$ with respect to parameters, shifts of the arguments, rescaling, etc. In the present paper, we develop a theory of Tannakian categories with semigroup actions, which will be used to attack such questions in full generality. Deligne studied actions of braid groups on categories and obtained a finite collection of axioms that characterizes such actions to apply it to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are finite free products of semigroups of the form $\mathbb{N}^n\times \mathbb{Z}/n_1\mathbb{Z}\times\ldots\times\mathbb{Z}/n_r\mathbb{Z}$ on Tannakian categories. This is the class of semigroups that appear in many applications., Comment: minor revision
- Published
- 2017
12. Isomorphisms of Twisted Hilbert Loop Algebras
- Author
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Timothée Marquis and Karl-Hermann Neeb
- Subjects
17B65, 17B70, 17B22, 17B10 ,General Mathematics ,010102 general mathematics ,Hilbert space ,Mathematics - Rings and Algebras ,01 natural sciences ,Combinatorics ,Loop (topology) ,symbols.namesake ,Isomorphism theorem ,Rings and Algebras (math.RA) ,Affine root system ,Product (mathematics) ,0103 physical sciences ,Lie algebra ,FOS: Mathematics ,symbols ,010307 mathematical physics ,Isomorphism ,Representation Theory (math.RT) ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Representation Theory ,Mathematics - Abstract
The closest infinite dimensional relatives of compact Lie algebras are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie algebra structure for which the scalar product is invariant. Locally affine Lie algebras (LALAs) correspond to double extensions of (twisted) loop algebras over simple Hilbert-Lie algebras $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$. They possess a root space decomposition whose corresponding root system is a locally affine root system of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a "minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard. In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$. The existence of such an isomorphism could also be derived from the classification of locally affine root systems, but for representation theoretic purposes it is crucial to obtain it explicitely as a deformation between two twists which is compatible with the root decompositions. We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$. In subsequent work, the present paper will be used to obtain a complete classification of the positive energy highest weight representations of affinisations of $\mathfrak{k}$., Comment: 22 pages; Minor corrections
- Published
- 2017
13. Ramification of the Eigencurve at Classical RM Points
- Author
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Adel Betina
- Subjects
Pure mathematics ,Mathematics - Number Theory ,General Mathematics ,Mathematics::Number Theory ,010102 general mathematics ,Modular form ,Local ring ,Weight space ,Subring ,Galois module ,01 natural sciences ,Base change ,Lift (mathematics) ,0103 physical sciences ,FOS: Mathematics ,Quadratic field ,010307 mathematical physics ,Number Theory (math.NT) ,0101 mathematics ,Mathematics - Abstract
J. Bellaïche and M. Dimitrov showed that the $p$-adic eigencurve is smooth but not étale over the weight space at $p$-regular theta series attached to a character of a real quadratic field $F$ in which $p$ splits. In this paper we prove the existence of an isomorphism between the subring fixed by the Atkin–Lehner involution of the completed local ring of the eigencurve at these points and a universal ring representing a pseudo-deformation problem. Additionally, we give a precise criterion for which the ramification index is exactly 2. We finish this paper by proving the smoothness of the nearly ordinary and ordinary Hecke algebras for Hilbert modular forms over $F$ at the overconvergent cuspidal Eisenstein points, being the base change lift for $\text{GL}(2)_{/F}$ of these theta series. Our approach uses deformations and pseudo-deformations of reducible Galois representations.
- Published
- 2019
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14. A Skolem–Mahler–Lech Theorem for Iterated Automorphisms of K–algebras
- Author
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Jeffrey C. Lagarias and Jason P. Bell
- Subjects
Pure mathematics ,Mathematics - Number Theory ,General Mathematics ,Algebraic extension ,Field (mathematics) ,Dynamical Systems (math.DS) ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,16. Peace & justice ,Automorphism ,Mathematics - Algebraic Geometry ,Skolem–Mahler–Lech theorem ,Scheme (mathematics) ,FOS: Mathematics ,Affine space ,Number Theory (math.NT) ,Mathematics - Dynamical Systems ,Primary: 11D45. Secondary: 14R10. 11Y55, 11D88 ,Algebra over a field ,Algebraic Geometry (math.AG) ,Finite set ,Mathematics - Abstract
This paper proves a commutative algebraic extension of a generalized Skolem-Mahler-Lech theorem due to the first author. Let $A$ be a finitely generated commutative $K$-algebra over a field of characteristic $0$, and let $\sigma$ be a $K$-algebra automorphism of $A$. Given ideals $I$ and $J$ of $A$, we show that the set $S$ of integers $m$ such that $\sigma^m(I) \supseteq J$ is a finite union of complete doubly infinite arithmetic progressions in $m$, up to the addition of a finite set. Alternatively, this result states that for an affine scheme $X$ of finite type over $K$, an automorphism $\sigma \in {\rm Aut}_K(X)$, and $Y$ and $Z$ any two closed subschemes of $X$, the set of integers $m$ with $\sigma^m(Z ) \subseteq Y$ is as above. The paper presents examples showing that this result may fail to hold if the affine scheme $X$ is not of finite type, or if $X$ is of finite type but the field $K$ has positive characteristic., Comment: 29 pages; to appear in the Canadian Journal of Mathematics
- Published
- 2015
15. Weighted Carleson Measure Spaces Associated with Different Homogeneities
- Author
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Xinfeng Wu
- Subjects
Carleson measure ,Pure mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we introduce weighted Carleson measure spaces associated with different homogeneities and prove that these spaces are the dual spaces of weighted Hardy spaces studied in a forthcoming paper. As an application, we establish the boundedness of composition of two Calderón–Zygmund operators with different homogeneities on the weighted Carleson measure spaces; this, in particular, provides the weighted endpoint estimates for the operators studied by Phong–Stein.
- Published
- 2014
16. Existence of Taut Foliations on Seifert Fibered Homology 3-spheres
- Author
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Shanti Caillat-Gibert and Daniel Matignon
- Subjects
Pure mathematics ,General Mathematics ,Taut foliation ,General Topology (math.GN) ,Physics::Physics Education ,Fibered knot ,Geometric Topology (math.GT) ,Homology (mathematics) ,Mathematics::Geometric Topology ,Mathematics - Geometric Topology ,FOS: Mathematics ,Mathematics::Differential Geometry ,Mathematics::Symplectic Geometry ,Mathematics - General Topology ,Mathematics - Abstract
This paper concerns the problem of existence of taut foliations among 3-manifolds. Since the contribution of David Gabai, we know that closed 3-manifolds with non-trivial second homology group admit a taut foliations. The essential part of this paper focuses on Seifert fibered homology 3-spheres. The result is quite different if they are integral or rational but non-integral homology 3-spheres. Concerning integral homology 3-spheres, we prove that all but the 3-sphere and the Poincar\'e 3-sphere admit a taut foliation. Concerning non-integral homology 3-spheres, we prove there are infinitely many which admit a taut foliation, and infinitely many without taut foliation. Moreover, we show that the geometries do not determine the existence of taut foliations on non-integral Seifert fibered homology 3-spheres., Comment: 34 pages, 1 figure
- Published
- 2014
17. Classic and Mirabolic Robinson–Schensted–Knuth Correspondence for Partial Flags
- Author
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Daniele Rosso
- Subjects
Mathematics::Combinatorics ,General Mathematics ,010102 general mathematics ,FLAGS register ,01 natural sciences ,Combinatorics ,Robinson–Schensted–Knuth correspondence ,Mathematics - Algebraic Geometry ,0103 physical sciences ,Line (geometry) ,FOS: Mathematics ,Mathematics - Combinatorics ,14M15 (Primary) 05A05 (Secondary) ,Combinatorics (math.CO) ,010307 mathematical physics ,0101 mathematics ,Variety (universal algebra) ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) ,Mathematics ,Flag (geometry) - Abstract
In this paper we first generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they lie, using the Robinson-Schensted-Knuth correspondence. Then we use this result to generalize the mirabolic Robinson-Schensted-Knuth correspondence defined by Travkin, to the case of two partial flags and a line., Comment: 27 pages, slightly rewritten to combine two papers into one and clarify some sections
- Published
- 2012
18. The Ample Cone for a K3 Surface
- Author
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Arthur Baragar
- Subjects
Surface (mathematics) ,Pure mathematics ,General Mathematics ,010102 general mathematics ,Lattice (group) ,Divisor (algebraic geometry) ,Algebraic number field ,01 natural sciences ,K3 surface ,Fractal ,Cone (topology) ,Hausdorff dimension ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In this paper, we give several pictorial fractal representations of the ample or Kahler cone for surfaces in a certain class of K3 surfaces. The class includes surfaces described by smooth (2, 2, 2) forms in P×P×P defined over a sufficiently large number field K, which have a line parallel to one of the axes, and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface’s group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be 1.296± .010. The ample cone or Kahler cone for a surface is a significant and often complicated geometric object. Though much is known about the ample cone, particularly for K3 surfaces, only a few non-trivial examples have been explicitly described. These include the ample cones with a finite number of sides (see [N1] for n = 3, and [N2, N3] for n ≥ 5; the case n = 4 is attributed to Vinberg in an unpublished work [N1]); the ample cone for a class of K3 surfaces with n = 3 [Ba3]; and the ample cones for several Kummer surfaces, which are K3 surfaces with n = 20 [V, K-K, Kon]. Though the complexity of the problem generically increases with n, the problem for K3 surfaces with maximal Picard number (n = 20) appear to be tractable because of the small size of the transcendental lattice. In this paper, we introduce accurate pictorial representations of the ample cone and the associated fractal for surfaces within a class of K3 surfaces with Picard number n = 4 (see Figures 1, 3, 4, and 5). As far as the author is aware, the associated fractal has not been studied in any great depth for any ample cone for which the fractal has a non-integer dimension, except the one in [Ba3]. The fractal in that case is Cantor-like (it is a subset of S) and rigorous bounds on its Hausdorff dimension are calculated in [Ba1]. The Hausdorff dimension of the fractal of this paper is estimated to be 1.296± .010. Our second main result is to relate the Hausdorff dimension of the fractal to the growth of the height of curves for an orbit of curves on a surface in this class. Precisely, let V be a surface within our class of K3 surfaces and let A = Aut(V/K) be its group of automorphisms over a sufficiently large number field K. Let D be an ample divisor on V and let C be a curve on V . Define NA(C)(t,D) = #{C′ ∈ A(C) : C′ ·D < t}. Here we have abused notation by letting C′ also represent the divisor class that contains C′. The intersection C′ ·D should be thought of as a logarithmic height of 2000 Mathematics Subject Classification. 14J28, 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05.
- Published
- 2011
19. A Variant of Lehmer’s Conjecture, II: The CM-case
- Author
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Sanoli Gun and V. Kumar Murty
- Subjects
General Mathematics ,010102 general mathematics ,Complex multiplication ,01 natural sciences ,Combinatorics ,Integer ,0103 physical sciences ,Eigenform ,Asymptotic formula ,010307 mathematical physics ,0101 mathematics ,Lehmer's conjecture ,Fourier series ,Mathematics - Abstract
Let f be a normalized Hecke eigenform with rational integer Fourier coefficients. It is an interesting question to know how often an integer n has a factor common with the n-th Fourier coefficient of f. It has been shown in previous papers that this happens very often. In this paper, we give an asymptotic formula for the number of integers n for which (n, a(n)) = 1, where a(n) is the n-th Fourier coefficient of a normalized Hecke eigenform f of weight 2 with rational integer Fourier coefficients and having complex multiplication.
- Published
- 2011
20. Locally Indecomposable Galois Representations
- Author
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Eknath Ghate and Vinayak Vatsal
- Subjects
Pure mathematics ,Galois cohomology ,General Mathematics ,Fundamental theorem of Galois theory ,010102 general mathematics ,Galois group ,Galois module ,01 natural sciences ,Normal basis ,Embedding problem ,symbols.namesake ,0103 physical sciences ,symbols ,010307 mathematical physics ,Galois extension ,0101 mathematics ,Indecomposable module ,Mathematics - Abstract
In a previous paper the authors showed that, under some technical conditions, the local Galois representations attached to the members of a non-CM family of ordinary cusp forms are indecomposable for all except possibly finitely many members of the family. In this paper we use deformation theoretic methods to give examples of non-CM families for which every classical member of weight at least two has a locally indecomposable Galois representation. School of Mathematics, Tata Institute of Fundamental Research, Mumbai 400005, India. e-mail: eghate@math.tifr.res.in Department of Mathematics, University of British Columbia, Vancouver, BC e-mail: vatsal@math.ubc.ca Received by the editors August 5, 2008. Published electronically December 29, 2010. AMS subject classification: 11F80. 1
- Published
- 2011
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