We discuss a class of bow varieties which can be viewed as Taub-NUT deformations of moduli spaces of instantons on noncommutative $mathbb R^4$. Via the generalized Legendre transform, we find the K"ahler potential on each of these spaces.<
{"title":"Deformations of Instanton Metrics","authors":"R. Bielawski, Yannic Borchard, Sergey A. Cherkis","doi":"10.3842/SIGMA.2023.041","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.041","url":null,"abstract":"We discuss a class of bow varieties which can be viewed as Taub-NUT deformations of moduli spaces of instantons on noncommutative $mathbb R^4$. Via the generalized Legendre transform, we find the K\"ahler potential on each of these spaces.<","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45080863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We introduce Stäckel separable coordinates on the covering manifolds $M_k$, where $k$ is a rational parameter, of certain constant-curvature Riemannian manifolds with the structure of warped manifold. These covering manifolds appear implicitly in literature as connected with superintegrable systems with polynomial in the momenta first integrals of arbitrarily high degree, such as the Tremblay-Turbiner-Winternitz system. We study here for the first time multiseparability and superintegrability of natural Hamiltonian systems on these manifolds and see how these properties depend on the parameter $k$.
{"title":"Separation of Variables and Superintegrability on Riemannian Coverings","authors":"C. Chanu, G. Rastelli","doi":"10.3842/SIGMA.2023.062","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.062","url":null,"abstract":"We introduce Stäckel separable coordinates on the covering manifolds $M_k$, where $k$ is a rational parameter, of certain constant-curvature Riemannian manifolds with the structure of warped manifold. These covering manifolds appear implicitly in literature as connected with superintegrable systems with polynomial in the momenta first integrals of arbitrarily high degree, such as the Tremblay-Turbiner-Winternitz system. We study here for the first time multiseparability and superintegrability of natural Hamiltonian systems on these manifolds and see how these properties depend on the parameter $k$.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42927117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide the law of large numbers for roots of finite free multiplicative convolution of polynomials which have only non-negative real roots. Moreover, we study the empirical root distributions of limit polynomials obtained through the law of large numbers of finite free multiplicative convolution when their degree tends to infinity.
{"title":"Law of Large Numbers for Roots of Finite Free Multiplicative Convolution of Polynomials","authors":"Katsunori Fujie, Yuki Ueda","doi":"10.3842/SIGMA.2023.004","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.004","url":null,"abstract":"We provide the law of large numbers for roots of finite free multiplicative convolution of polynomials which have only non-negative real roots. Moreover, we study the empirical root distributions of limit polynomials obtained through the law of large numbers of finite free multiplicative convolution when their degree tends to infinity.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41880305","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. Figueroa-O’Farrill, Ross Grassie, Stefan Prohazka
We exhibit all spatially isotropic homogeneous Galilean spacetimes of dimension (n+1)>=4, including the novel torsional ones, as null reductions of homogeneous pp-wave spacetimes. We also show that the pp-waves are sourced by pure radiation fields and analyse their global properties.
{"title":"From pp-Waves to Galilean Spacetimes","authors":"J. Figueroa-O’Farrill, Ross Grassie, Stefan Prohazka","doi":"10.3842/SIGMA.2023.035","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.035","url":null,"abstract":"We exhibit all spatially isotropic homogeneous Galilean spacetimes of dimension (n+1)>=4, including the novel torsional ones, as null reductions of homogeneous pp-wave spacetimes. We also show that the pp-waves are sourced by pure radiation fields and analyse their global properties.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43821037","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study tautological vector bundles over the Hilbert scheme of points on surfaces. For each K-trivial surface, we write down a simple criterion ensuring that the tautological bundles are big and nef, and illustrate it by examples. In the K3 case, we extend recent constructions and results of Bini, Boissi`ere and Flamini from the Hilbert scheme of 2 and 3 points to an arbitrary number of points. Among the K-trivial surfaces, the case of Enriques surfaces is the most involved. Our techniques apply to other smooth projective surfaces, including blowups of K3s and minimal surfaces of general type, as well as to the punctual Quot schemes of curves.
{"title":"Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points","authors":"D. Oprea","doi":"10.3842/SIGMA.2022.061","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.061","url":null,"abstract":"We study tautological vector bundles over the Hilbert scheme of points on surfaces. For each K-trivial surface, we write down a simple criterion ensuring that the tautological bundles are big and nef, and illustrate it by examples. In the K3 case, we extend recent constructions and results of Bini, Boissi`ere and Flamini from the Hilbert scheme of 2 and 3 points to an arbitrary number of points. Among the K-trivial surfaces, the case of Enriques surfaces is the most involved. Our techniques apply to other smooth projective surfaces, including blowups of K3s and minimal surfaces of general type, as well as to the punctual Quot schemes of curves.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45360549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct families of asymptotically locally hyperbolic Riemannian metrics with constant scalar curvature (i.e., time symmetric vacuum general relativistic initial data sets with negative cosmological constant), with prescribed topology of apparent horizons and of the conformal boundary at infinity, and with controlled mass. In particular we obtain new classes of solutions with negative mass.
{"title":"On Asymptotically Locally Hyperbolic Metrics with Negative Mass","authors":"Piotr T. Chru'sciel, E. Delay","doi":"10.3842/SIGMA.2023.005","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.005","url":null,"abstract":"We construct families of asymptotically locally hyperbolic Riemannian metrics with constant scalar curvature (i.e., time symmetric vacuum general relativistic initial data sets with negative cosmological constant), with prescribed topology of apparent horizons and of the conformal boundary at infinity, and with controlled mass. In particular we obtain new classes of solutions with negative mass.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43762162","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the space of (orthogonal) almost complex structures on closed six-dimensional manifolds as the space of sections of the twistor space for a given metric. For a connected six-manifold with vanishing first Betti number, we express the space of almost complex structures as a quotient of the space of sections of a seven-sphere bundle over the manifold by a circle action, and then use this description to compute the rational homotopy theoretic minimal model of the components that satisfy a certain Chern number condition. We further obtain a formula for the homological intersection number of two sections of the twistor space in terms of the Chern classes of the corresponding almost complex structures.
{"title":"Topology of Almost Complex Structures on Six-Manifolds","authors":"Gustavo Granja, A. Milivojević","doi":"10.3842/SIGMA.2022.093","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.093","url":null,"abstract":"We study the space of (orthogonal) almost complex structures on closed six-dimensional manifolds as the space of sections of the twistor space for a given metric. For a connected six-manifold with vanishing first Betti number, we express the space of almost complex structures as a quotient of the space of sections of a seven-sphere bundle over the manifold by a circle action, and then use this description to compute the rational homotopy theoretic minimal model of the components that satisfy a certain Chern number condition. We further obtain a formula for the homological intersection number of two sections of the twistor space in terms of the Chern classes of the corresponding almost complex structures.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46336898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $(G,G_1)=(G,(G^sigma)_0)$ be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces $D_1=G_1/K_1subset D=G/K$, realized as bounded symmetric domains in complex vector spaces ${mathfrak p}^+_1:=({mathfrak p}^+)^sigmasubset{mathfrak p}^+$ respectively. Then the universal covering group $widetilde{G}$ of $G$ acts unitarily on the weighted Bergman space ${mathcal H}_lambda(D)subset{mathcal O}(D)={mathcal O}_lambda(D)$ on $D$ for sufficiently large $lambda$. Its restriction to the subgroup $widetilde{G}_1$ decomposes discretely and multiplicity-freely, and its branching law is given explicitly by Hua-Kostant-Schmid-Kobayashi's formula in terms of the $widetilde{K}_1$-decomposition of the space ${mathcal P}({mathfrak p}^+_2)$ of polynomials on ${mathfrak p}^+_2:=({mathfrak p}^+)^{-sigma}subset{mathfrak p}^+$. The object of this article is to understand the decomposition of the restriction ${mathcal H}_lambda(D)|_{widetilde{G}_1}$ by studying the weighted Bergman inner product on each $widetilde{K}_1$-type in ${mathcal P}({mathfrak p}^+_2)subset{mathcal H}_lambda(D)$. For example, by computing explicitly the norm $Vert fVert_lambda$ for $f=f(x_2)in{mathcal P}({mathfrak p}^+_2)$, we can determine the Parseval-Plancherel-type formula for the decomposition of ${mathcal H}_lambda(D)|_{widetilde{G}_1}$. Also, by computing the poles of $langle f(x_2),{rm e}^{(x|overline{z})_{{mathfrak p}^+}}rangle_{lambda,x}$ for $f(x_2)in{mathcal P}({mathfrak p}^+_2)$, $x=(x_1,x_2)$, $zin{mathfrak p}^+={mathfrak p}^+_1oplus{mathfrak p}^+_2$, we can get some information on branching of ${mathcal O}_lambda(D)|_{widetilde{G}_1}$ also for $lambda$ in non-unitary range. In this article we consider these problems for all $widetilde{K}_1$-types in ${mathcal P}({mathfrak p}^+_2)$.
{"title":"Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Parseval-Plancherel-Type Formulas under Subgroups","authors":"Ryosuke Nakahama","doi":"10.3842/SIGMA.2023.049","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.049","url":null,"abstract":"Let $(G,G_1)=(G,(G^sigma)_0)$ be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces $D_1=G_1/K_1subset D=G/K$, realized as bounded symmetric domains in complex vector spaces ${mathfrak p}^+_1:=({mathfrak p}^+)^sigmasubset{mathfrak p}^+$ respectively. Then the universal covering group $widetilde{G}$ of $G$ acts unitarily on the weighted Bergman space ${mathcal H}_lambda(D)subset{mathcal O}(D)={mathcal O}_lambda(D)$ on $D$ for sufficiently large $lambda$. Its restriction to the subgroup $widetilde{G}_1$ decomposes discretely and multiplicity-freely, and its branching law is given explicitly by Hua-Kostant-Schmid-Kobayashi's formula in terms of the $widetilde{K}_1$-decomposition of the space ${mathcal P}({mathfrak p}^+_2)$ of polynomials on ${mathfrak p}^+_2:=({mathfrak p}^+)^{-sigma}subset{mathfrak p}^+$. The object of this article is to understand the decomposition of the restriction ${mathcal H}_lambda(D)|_{widetilde{G}_1}$ by studying the weighted Bergman inner product on each $widetilde{K}_1$-type in ${mathcal P}({mathfrak p}^+_2)subset{mathcal H}_lambda(D)$. For example, by computing explicitly the norm $Vert fVert_lambda$ for $f=f(x_2)in{mathcal P}({mathfrak p}^+_2)$, we can determine the Parseval-Plancherel-type formula for the decomposition of ${mathcal H}_lambda(D)|_{widetilde{G}_1}$. Also, by computing the poles of $langle f(x_2),{rm e}^{(x|overline{z})_{{mathfrak p}^+}}rangle_{lambda,x}$ for $f(x_2)in{mathcal P}({mathfrak p}^+_2)$, $x=(x_1,x_2)$, $zin{mathfrak p}^+={mathfrak p}^+_1oplus{mathfrak p}^+_2$, we can get some information on branching of ${mathcal O}_lambda(D)|_{widetilde{G}_1}$ also for $lambda$ in non-unitary range. In this article we consider these problems for all $widetilde{K}_1$-types in ${mathcal P}({mathfrak p}^+_2)$.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47841101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In Book 1, Proposition 7, Problem 2 of his 1687 Philosophiae Naturalis Principia Mathematica, Isaac Newton poses and answers the following question: Let the orbit of a particle moving in a central force field be an off-center circle. How does the magnitude of the force depend on the position of the particle onthat circle? In this article, we identify a potential that can produce such a force, only at zero energy. We further map the zero-energy orbits in this potential to finite-energy free motion orbits on a sphere; such a duality is a particular instance of a general result by Goursat, from 1887. The map itself is an inverse stereographic projection, and this fact explains the circularity of the zero-energy orbits in the system of interest. Finally, we identify an additional integral of motion - an analogue of the Runge-Lenz vector in the Coulomb problem - that is responsible for the closeness of the zero-energy orbits in our problem.
{"title":"A Novel Potential Featuring Off-Center Circular Orbits","authors":"M. Olshanii","doi":"10.3842/SIGMA.2023.001","DOIUrl":"https://doi.org/10.3842/SIGMA.2023.001","url":null,"abstract":"In Book 1, Proposition 7, Problem 2 of his 1687 Philosophiae Naturalis Principia Mathematica, Isaac Newton poses and answers the following question: Let the orbit of a particle moving in a central force field be an off-center circle. How does the magnitude of the force depend on the position of the particle onthat circle? In this article, we identify a potential that can produce such a force, only at zero energy. We further map the zero-energy orbits in this potential to finite-energy free motion orbits on a sphere; such a duality is a particular instance of a general result by Goursat, from 1887. The map itself is an inverse stereographic projection, and this fact explains the circularity of the zero-energy orbits in the system of interest. Finally, we identify an additional integral of motion - an analogue of the Runge-Lenz vector in the Coulomb problem - that is responsible for the closeness of the zero-energy orbits in our problem.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48616665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Weierstrass curve is a pointed curve $(X,infty)$ with a numerical semigroup $H_X$, which is a normalization of the curve given by the Weierstrass canonical form, $y^r + A_{1}(x) y^{r-1} + A_{2}(x) y^{r-2} +dots + A_{r-1}(x) y + A_{r}(x)=0$ where each $A_j$ is a polynomial in $x$ of degree $leq j s/r$ for certain coprime positive integers $r$ and $s$, $r$<$s$, such that the generators of the Weierstrass non-gap sequence $H_X$ at $infty$ include $r$ and $s$. The Weierstrass curve has the projection $varpi_rcolon X to {mathbb P}$, $(x,y)mapsto x$, as a covering space. Let $R_X := {mathbf H}^0(X, {mathcal O}_X(*infty))$ and $R_{mathbb P} := {mathbf H}^0({mathbb P}, {mathcal O}_{mathbb P}(*infty))$ whose affine part is ${mathbb C}[x]$. In this paper, for every Weierstrass curve $X$, we show the explicit expression of the complementary module $R_X^{mathfrak c}$ of $R_{mathbb P}$-module $R_X$ as an extension of the expression of the plane Weierstrass curves by Kunz. The extension naturally leads the explicit expressions of the holomorphic one form except $infty$, ${mathbf H}^0({mathbb P}, {mathcal A}_{mathbb P}(*infty))$ in terms of $R_X$. Since for every compact Riemann surface, we find a Weierstrass curve that is bi-rational to the surface, we also comment that the explicit expression of $R_X^{mathfrak c}$ naturally leads the algebraic construction of generalized Weierstrass' sigma functions for every compact Riemann surface and is also connected with the data on how the Riemann surface is embedded into the universal Grassmannian manifolds.
{"title":"Complementary Modules of Weierstrass Canonical Forms","authors":"J. Komeda, S. Matsutani, E. Previato","doi":"10.3842/SIGMA.2022.098","DOIUrl":"https://doi.org/10.3842/SIGMA.2022.098","url":null,"abstract":"The Weierstrass curve is a pointed curve $(X,infty)$ with a numerical semigroup $H_X$, which is a normalization of the curve given by the Weierstrass canonical form, $y^r + A_{1}(x) y^{r-1} + A_{2}(x) y^{r-2} +dots + A_{r-1}(x) y + A_{r}(x)=0$ where each $A_j$ is a polynomial in $x$ of degree $leq j s/r$ for certain coprime positive integers $r$ and $s$, $r$<$s$, such that the generators of the Weierstrass non-gap sequence $H_X$ at $infty$ include $r$ and $s$. The Weierstrass curve has the projection $varpi_rcolon X to {mathbb P}$, $(x,y)mapsto x$, as a covering space. Let $R_X := {mathbf H}^0(X, {mathcal O}_X(*infty))$ and $R_{mathbb P} := {mathbf H}^0({mathbb P}, {mathcal O}_{mathbb P}(*infty))$ whose affine part is ${mathbb C}[x]$. In this paper, for every Weierstrass curve $X$, we show the explicit expression of the complementary module $R_X^{mathfrak c}$ of $R_{mathbb P}$-module $R_X$ as an extension of the expression of the plane Weierstrass curves by Kunz. The extension naturally leads the explicit expressions of the holomorphic one form except $infty$, ${mathbf H}^0({mathbb P}, {mathcal A}_{mathbb P}(*infty))$ in terms of $R_X$. Since for every compact Riemann surface, we find a Weierstrass curve that is bi-rational to the surface, we also comment that the explicit expression of $R_X^{mathfrak c}$ naturally leads the algebraic construction of generalized Weierstrass' sigma functions for every compact Riemann surface and is also connected with the data on how the Riemann surface is embedded into the universal Grassmannian manifolds.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48355842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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