Se agregan sesiones hasta Analisis no lineal en espacios de Banach
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Germán Correa
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mail: viviana.diaz@uns.edu.ar
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- nombre: Edith Padrón Fernandez
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mail: mepadron@ull.es
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charlas:
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- titulo: Conserved quantities and the existence of (twisted) Poisson brackets in nonholonomic mechanics
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In this talk, I will start by explaining the nonhamiltonian nature of nonholonomic systems, and then we will study the "hamiltonization problem" from a geometric standpoint. By making use of symmetries and suitable first integrals of the system, we will explicitly define a new bracket on the reduced space codifying the nonholonomic dynamics that, in many examples, is a genuine Poisson bracket making the system hamiltonian (in general, the new bracket carries an almost symplectic foliation determined by the first integrals). This is a joint work with Luis Yapu-Quispe.
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speaker: Paula Balseiro (Universidad Federal Fluminense, Brasil)
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- titulo: From retraction maps to geometric integrators for optimal control problems
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Retraction maps are used in many research fields such as approximation of trajectories of differential equations, optimization theory, interpolation theory etc. In this talk we will review the concept of retraction map in differentiable manifolds to generalize it to obtain an extended retraction map from the tangent bundle of the configuration manifold to two copies of this manifold. After suitably lifting the new retraction map to the cotangent bundle, the typical phase space for Hamiltonian mechanical systems, we will be able to define geometric integrators for optimal control problems. This is a joint work with David Martín de Diego.
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speaker: María Barbero Liñán (Universidad Politécnica de Madrid, España)
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- titulo: Relación entre la teoría de campos de Chern-Simons y relatividad general en dimensión \(3\), desde el punto de vista de los problemas variacionales de Griffiths
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La teoría de campos de Chern-Simons es una teoría de campos topológica, y puede formularse sobre cualquier fibrado principal cuyo grupo de estructura admita una forma bilineal invariante. Consideremos un fibrado principal \(\pi:P\to M\) con grupo de estructura \(G\) y sea \(K\subset G\) un subgrupo; tomemos la familia de problemas variacionales de Chern-Simons sobre todos los subfibrados de \(P\) con fibra \(K\). En la presente charla explicaremos cómo esta familia de problemas variacionales puede codificarse mediante un único principio variacional de tipo Griffiths. Finalmente, utilizaremos el problema variacional construído para entender desde un punto de vista geométrico la correspondencia entre gravedad en dimensión \(2+1\) y la teoría de Chern-Simons.
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speaker: Santiago Capriotti (Universidad Nacional del Sur, Argentina)
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- titulo: Lie group's exponential curves and the Hamilton-Jacobi theory
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In this talk we present an extended version of the Hamilton-Jacobi equation (HJE), valid for general dynamical systems defined by vector fields (not only by the Hamiltonian ones), and a result which ensures that, if we have a complete solution of the HJE for a given dynamical system, inside a certain subclass of systems, then such a system can be integrated up to quadratures. Then we apply this result to show that the exponential curves \(\exp\left(\eta\,t\right)\) of any Lie group, which are the integral curves of the left invariant vector fields in the group, can be constructed up to quadratures (unless for certain elements \(\eta\) inside its corresponding Lie algebra). This gives rise to an alternative concrete expression of \(\exp\left(\eta\,t\right)\), different to those that appears in the literature for matrix Lie groups.
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speaker: Sergio Grillo (Centro Atómico Bariloche, Argentina)
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- titulo: Null hyperpolygons and quasi-parabolic Higgs bundles
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Hyperpolygons spaces are a family of hyperkähler manifolds that can be obtained from coadjoint orbits by hyperkähler reduction. Jointly with L. Godinho, we showed that these spaces are isomorphic to certain families of parabolic Higgs bundles, when a suitable condition between the parabolic weights and the spectra of the coadjoint orbits is satisfied. In analogy to this construction, we introduce two moduli spaces: the moduli spaces of quasi-parabolic \(SL(2,\mathbb{C})\)-Higgs bundles over \(\mathbb{C}\mathbb{P}^1\) on one hand and the null hyperpolygon spaces on the other, and establish an isomorphism between them. Finally we describe the fixed loci of natural involutions defined on these spaces and relate them to the moduli space of null hyperpolygons in the Minkowski 3-space. This is based on joint works with Leonor Godinho.
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speaker: Alessia Mandini (Pontifícia Universidade Católica do Rio de Janeiro, Brasil)
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- titulo: Kinetic nonholonomic dynamics is neither Hamiltonian nor variational and however...
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It is well-known that kinetic nonholonomic dynamics is neither Hamiltonian nor variational. However, in this talk, after introducing a geometrical description of nonholonomic dynamics, I will present a result which is a little surprising. Namely, for a kinetic nonholonomic system and a given point \(q\) of the configuration space \(Q\), one may define:
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<ul>
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<li> A submanifold \({\mathcal M}^{nh}_q\) of \(Q\), which contains to the point \(q\) and whose dimension is equal to the rank of the constraint distribution, and</li>
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<li> A family of Riemannian metrics on \({\mathcal M}^{nh}_q\) such that the geodesics of every one of these metrics with starting point \(q\) are just the nonholonomic trajectories with the same starting point \(q\).</li>
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</ul>
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In particular, the previous facts imply that the kinetic nonholonomic trajectories with starting point \(q\), for sufficiently small times, minimize length in \({\mathcal M}^{nh}_q\)!
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speaker: Juan Carlos Marrero (Universidad de La Laguna, España)
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- titulo: Discrete variational calculus and accelerated methods in optimization
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Many of the new developments in machine learning are connected with gradient-based optimization methods. Recently, these methods have been studied using a variational perspective. This has opened up the possibility of introducing variational and symplectic integration methods using geometric integrators. In particular, in thistle, we introduce variational integrators which allows us to derive different methods for optimization. However, since the systems are explicitly time-dependent, the preservation of the symplecticity property occurs solely on the fibers. Finally, using discrete Lagrange-d'Alembert principle we produce optimization methods whose behavior is similar to the classical Nesterov method reducing the oscillations of typical momentum methods. Joint work with Cédric M. Campos and Alejandro Mahillo.
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speaker: David Martín de Diego (Instituto de Ciencias Matemática, España)
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- titulo: Sobre sistemas mecánicos discretos forzados y la reducción de Routh discreta
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Los sistemas lagrangianos y hamiltonianos con fuerzas aparecen en distintos contextos tales como sistemas con fuerzas de control o sistemas con fuerzas de disipación y fricción. También es usual que los sistemas dinámicos que se obtienen al reducir una simetría de un sistema mecánico sean sistemas forzados. En los casos en los que las fuerzas no puedan ser absorbidas por el lagrangiano o el hamiltoniano como parte de un potencial, se realiza una descripción variacional alternativa del caso sin fuerzas. Como en el caso continuo, cuando se realiza un proceso de reducción de una simetría de un sistema mecánico discreto, es usual que el sistema dinámico reducido presentes términos que se pueden interpretar como fuerzas. Entre otras razones, esto hace que resulte interesante estudiar las características de los sistemas discretos forzados. Un caso particularmente interesante de reducción de simetrías de un sistema mecánico es el que se basa en el uso de sus cantidades conservadas. Cuando se considera un sistema mecánico discreto que admite una aplicación momento que se conserva sobre las trayectorias del sistema, se aplica el proceso conocido como la reducción de Routh discreta que da lugar a un sistema forzado. En esta charla se consideran sistemas mecánicos discretos forzados. A partir de su dinámica definida por un principio de Lagrange d'Alembert discreto que se define modificando convenientemente el principio variacional usual, se estudia la existencia de estructuras simplécticas y su posible conservación por la evolución del sistema. En particular, se considera el proceso de reducción de Routh discreta para una simetría no necesariamente abeliana de un sistema mecánico discreto como un caso particular de un proceso de reducción de simetrías más general. También se analizan la existencia y la conservación de estructuras simplécticas sobre el espacio reducido en el marco de los sistemas forzados teniendo en cuenta las características propias de este caso especial.
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speaker: Marcela Zuccalli (Universidad Nacional de La Plata, Argentina)
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- sesion: Teoría de códigos y temas afines
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organizadores:
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- nombre: Ricardo Podestá
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mail: podesta@famaf.unc.edu.ar
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- nombre: Claudio Qureshi
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mail: cqureshi@gmail.com
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charlas:
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- titulo: The conorm code of an AG-code
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Let \(\mathbb{F}_q\) be a finite field with \(q\) elements. For a given trascendental element \(x\) over \(\mathbb{F}_q\), the field of fractions of the ring \(\mathbb{F}_q[x]\) is denoted as \(\mathbb{F}_q(x)\) and it is called a rational function field over \(\mathbb{F}_q\). An (algebraic) function field \(F\) of one variable over \(\mathbb{F}_q\) is a field extension \(F/\mathbb{F}_q(x)\) of finite degree. The \textit{Riemann-Roch space} associated to a divisor \(G\) of \(F\) is the vector space over \(\mathbb{F}_q\) defined as
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$$\mathcal{L}(G)=\{x\in F\,:\, (x)\geq G\}\cup \{0\},$$
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where \((x)\) denotes the principal divisor of \(x\). It turns out that \(\mathcal{L}(G)\) is a finite dimensional vector space over \(\mathbb{F}_q\) for any divisor \(G\) of \(F\). Given disjoint divisors \(D=P_1+\cdots+P_n\) and \(G\) of \(F/\mathbb{F}_q\), where \(P_1,\ldots,P_n\) are different rational places, the \textit{algebraic geometry code} (AG-code for short) associated to \(D\) and \(G\) is defined as
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$$C_\mathcal{L}^F (D,G) = \{(x(P_1),\ldots, x(P_n))\,:\,x\in \mathcal{L}(G)\}\subseteq (\mathbb{F}_q)^n,$$
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where \(x(P_i)\) denotes the residue class of \(x\) modulo \(P_i\) for \(i=1,\ldots,n\).
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In this talk the concept of the <em>conorm code,/em> associated to an AG-code will be introduced. We will show some interesting properties of this new code since some well known families of codes such as repetition codes, Hermitian codes and Reed-Solomon codes can be obtained as conorm codes from other more basic codes. We will see that in some particular cases over geometric Galois extensions of function fields, the conorm code and the original code are different representations of the same algebraic geometry code.
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speaker: María Chara (Universidad Nacional del Litoral, Argentina)
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- titulo: Códigos Reticulados
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Reticulados são subconjuntos discretos do espaço euclidiano n dimensional gerados por combinações inteiras de um conjunto de vetores independentes. Reticulados vem sendo usados em processos de codificação para transmissão de sinais particularmente para comunicações do tipo MIMO (multiple-input/multiple-output) e também em esquemas criptográficos na recente área da chamada criptografia pós-quântica. Nesta apresentação faremos uma abordagem geral deste tema e apresentaremos de forma resumida alguns resultados recentes sobre códigos reticulados perfeitos em diferentes métricas, constelações de Voronoi em reticulados construídos a partir de códigos em anéis finitos e uma extensão do problema ``Ring-LWE'' de criptografia baseada em reticulados que inclui uma classe mais geral de reticulados algébricos.
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speaker: Sueli I. R. Costa (Universidade Estadual de Campinas, Brasil)
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- titulo: The Generalized Covering Radii of Codes
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The <em>generalized covering-radius hierarchy</em> of a linear code is a new property of codes which was motivated by an application to database linear querying, such as private information-retrieval protocols. It characterizes the trade-off between storage amount, latency, and access complexity, in such database systems.
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In this work, we introduce and discuss three definitions for the generalized covering radius of a code, highlighting the combinatorial, geometric, and algebraic properties of this concept and prove them to be equivalent. We also discuss a connection between the generalized covering radii and the generalized Hamming weights of codes by showing that the latter is in fact a packing problem with some rank relaxation.
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Other contributions to the subject include bounds relating various parameters of codes to the generalized covering radii and an asymptotic upper bound (for particular parameters) which shows that the generalized covering radii improves the naive approach.
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Joint work with Dor Elimelech and Moshe Schwartz (Ben-Gurion University/Israel).
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speaker: Marcelo Firer (Universidade Estadual de Campinas, Brasil)
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- titulo: A novel version of group codes
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Let \(G\) be a finite abelian group, with \(G=\prod_{i=1}^n \langle g_{i} \rangle \) where \(|\langle g_{i} \rangle|=m_{i}\). Then every element in \(G\) can be uniquely written as \(\prod_{i=1}^n g_i^{\epsilon_i}\) where \(0\leq \epsilon_i \leq m_{i}-1 \). To determine a measure of the separation between two elements of \(G\) we use the \texttt{Minkowski distance \(l_1\)}, which is given by
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$$l_{1}\Big(\prod_{i=1}^n g_i^{\epsilon_i}, \prod_{i=1}^n g_i^{\delta_i}\Big) = \sum_{i=1}^n |\epsilon_i-\delta_i|.$$
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A <em>grid code</em> \(\codeC\) is a subset of \(G\), if \(\codeC\) is subgroup of \(G\), then it is said that \(\codeC\) is a \texttt{group code}. The elements of \(\codeC\) are called \texttt{codewords}. The \texttt{minimum distance} \(d\) of a code \(\codeC\) is defined as usually, that is, as the smallest distance between any two different elements of \(\codeC\). Let \(\codeC\) be a code of \(G\) with minimum distance \(d\). Then we say that \(\codeC\) is a \((n,|\codeC|,d)\)-code over \(G\) and \((n,|\codeC|,d)\) are its parameters.
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In this talk, we consider such codes, and we prove some classical results on block codes, like Singleton Bound and others.
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speaker: Ismael Gutiérrez (Universidad del Norte, Colombia)
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- titulo: Direct sum of Barnes-Wall lattices via totally real number fields
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Let \(\mathbb{K}\) be a number field of degree \(n\), \(\mathcal O_{\mathbb{K}}\) its ring of integers and \(\alpha \in \mathcal O_{\mathbb{K}}\) a totally real and totally positive element. In 1999, Eva Bayer-Fluckiger introduced a twisted embedding \(\sigma_{\alpha}:\mathbb{K} \rightarrow \mathbb{R}^{n}\) such that if \(\mathcal I \subseteq \mathcal O_{\mathbb{K}}\) is a free \(\mathbb{Z}\)-module of rank \(n\), then \(\sigma_{\alpha}(\mathcal I)\) is a lattice in \(\mathbb{R}^{n}\). It was shown that if \(\mathbb{K}\) is a totally real number field, then \(\sigma_{\alpha}(\mathcal I)\) is a full diversity lattice. In this talk we will approach constructions of direct sum of Barnes-Wall lattices \(BW_n\) for \(n=4,8\) and \(16\) via ideals of the ring of the integers \(\mathbb{Z}[\zeta_{2^{r}q} + \zeta_{2^{r}q}^{-1}]\) for \(q = 3, 5\) and \(15\). Our focus is on totally real number fields since the associated lattices have full diversity and then may be suitable for signal transmission over both Gaussian and Rayleigh fading channels. The minimum product distances of such constructions are also presented here. This is a joint work with Jo\~{a}o E. Strapasson, Agnaldo J. Ferrari and Sueli I. R. Costa and it was partially supported by Fapesp 2013/25977-7, 2014/14449-2 and 2015/17167-0 and CNPq 432735/2016-0 and 429346/2018-2.
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speaker: Grasiele C. Jorge (Universidade Federal de São Paulo, Brasil)
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- titulo: AG codes, bases of Riemann-Roch spaces and Weierstrass semigroups
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In this talk we are going to discuss some results on the explicit construction of bases of Riemann-Roch spaces, weierstrass semigroups and the floor of certain divisors. We also will apply these results to get AG codes with good parameters.
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speaker: Horacio Navarro (Universidad del Valle, Colombia)
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- titulo: The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs
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We use known characterizations of generalized Paley graphs which are Cartesian decomposable to explicitly compute the spectra of the corresponding associated irreducible cyclic codes. As applications, we give reduction formulas for the number of rational points in Artin-Schreier curves defined over extension fields. This talk is based on a recent joint work with Ricardo Podestá.
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speaker: Denis Videla (Universidad Nacional de Córdoba, Argentina)
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- titulo: A Decoding Algorithm of MDS Array Codes
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This work aims to present a construction of MDS (Maximum Distance Separable) array codes constructed by using superregular matrices, in particular Vandermonde matrices and Cauchy matrices, and the Frobenius companion matrix obtained through a primitive polynomial over \(\mathbb{F}_q[x]\). Array codes are two-dimensional error correction codes whose main characteristic is the ability to correct burst of errors, that is, errors that occur in consecutive bits. MDS codes are codes in which the minimum distance is the maximum possible. This characteristic is important because, in coding theory, the minimum distance is related to the error correction capacity of the code in addition to providing maximum protection against failures of a device for a given amount of redundancy. Based on this construction, a decoding algorithm is presented to correct up to two bursts of errors in MDS array codes with parameters \([m + k, k, m + 1]\) over \(\mathbb{F}_q^b\), for all \(m \geq 4\), where \(b\) refers to the length of the error. In addition, some examples are presented to correct three bursts of errors. This is a joint work with Débora Beatriz Claro Zanitti, São Paulo State University (UNESP).
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speaker: Cintya Wink de Oliveira Benedito (Universidade Estadual Paulista, Brasil)
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- sesion: Análisis no lineal en espacios de Banach
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organizadores:
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- nombre: Gerardo Botelho
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