Marius Mitrea's Books

Multi-Layer Potentials and Boundary Problems

for Higher-Order Elliptic Systems in Lipschitz Domains

By: Irina Mitrea,Marius Mitrea

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Multi-Layer Potentials and Boundary Problems

for Higher-Order Elliptic Systems in Lipschitz Domains

By: Irina Mitrea,Marius Mitrea

Many phenomena in engineering and mathematical physics can be modeled by means of boundary value problems for a certain elliptic differential operator in a given domain. When the differential operator under discussion is of second order a variety of tools are available for dealing with such problems, including boundary integral methods, variational methods, harmonic measure techniques, and methods based on classical harmonic analysis. When the differential operator is of higher-order (as is the case, e.g., with anisotropic plate bending when one deals with a fourth order operator) only a few options could be successfully implemented. In the 1970s Alberto Calderón, one of the founders of the modern theory of Singular Integral Operators, advocated the use of layer potentials for the treatment of higher-order elliptic boundary value problems. The present monograph represents the first systematic treatment based on this approach. This research monograph lays, for the first time, the mathematical foundation aimed at solving boundary value problems for higher-order elliptic operators in non-smooth domains using the layer potential method and addresses a comprehensive range of topics, dealing with elliptic boundary value problems in non-smooth domains including layer potentials, jump relations, non-tangential maximal function estimates, multi-traces and extensions, boundary value problems with data in Whitney–Lebesque spaces, Whitney–Besov spaces, Whitney–Sobolev- based Lebesgue spaces, Whitney–Triebel–Lizorkin spaces,Whitney–Sobolev-based Hardy spaces, Whitney–BMO and Whitney–VMO spaces.

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430

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Springer

Published Date

ISBN 10

3642326668

ISBN 13

9783642326660

Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds

By: Dorina Mitrea,Marius Mitrea,Michael Taylor

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Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds

By: Dorina Mitrea,Marius Mitrea,Michael Taylor

The general aim of the present monograph is to study boundary-value problems for second-order elliptic operators in Lipschitz sub domains of Riemannian manifolds. In the first part (ss1-4), we develop a theory for Cauchy type operators on Lipschitz submanifolds of co dimension one (focused on boundedness properties and jump relations) and solve the $Lp$-Dirichlet problem, with $p$ close to $2$, for general second-order strongly elliptic systems. The solution is represented in the form of layer potentials and optimal non tangential maximal function estimates are established.This analysis is carried out under smoothness assumptions (for the coefficients of the operator, metric tensor and the underlying domain) which are in the nature of best possible. In the second part of the monograph, ss5-13, we further specialize this discussion to the case of Hodge Laplacian $\Delta: =-d\delta-\delta d$. This time, the goal is to identify all (pairs of) natural boundary conditions of Neumann type. Owing to the structural richness of the higher degree case we are considering, the theory developed here encompasses in a unitary fashion many basic PDE's of mathematical physics. Its scope extends to also cover Maxwell's equations, dealt with separately in s14. The main tools are those of PDE's and harmonic analysis, occasionally supplemented with some basic facts from algebraic topology and differential geometry.

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137

Publisher

American Mathematical Soc.

Published Date

ISBN 10

082182659X

ISBN 13

9780821826591

Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces

A Sharp Theory

By: Ryan Alvarado,Marius Mitrea

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Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces

A Sharp Theory

By: Ryan Alvarado,Marius Mitrea

Systematically constructing an optimal theory, this monograph develops and explores several approaches to Hardy spaces in the setting of Alhlfors-regular quasi-metric spaces. The text is divided into two main parts, with the first part providing atomic, molecular, and grand maximal function characterizations of Hardy spaces and formulates sharp versions of basic analytical tools for quasi-metric spaces, such as a Lebesgue differentiation theorem with minimal demands on the underlying measure, a maximally smooth approximation to the identity and a Calderon-Zygmund decomposition for distributions. These results are of independent interest. The second part establishes very general criteria guaranteeing that a linear operator acts continuously from a Hardy space into a topological vector space, emphasizing the role of the action of the operator on atoms. Applications include the solvability of the Dirichlet problem for elliptic systems in the upper-half space with boundary data from Hardy spaces. The tools established in the first part are then used to develop a sharp theory of Besov and Triebel-Lizorkin spaces in Ahlfors-regular quasi-metric spaces. The monograph is largely self-contained and is intended for mathematicians, graduate students and professionals with a mathematical background who are interested in the interplay between analysis and geometry.

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491

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Springer

Published Date

ISBN 10

3319181327

ISBN 13

9783319181325

Hypercontractivity in Group von Neumann Algebras

By: Marius Junge,Carlos Palazuelos,Javier Parcet,Mathilde Perrin

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Hypercontractivity in Group von Neumann Algebras

By: Marius Junge,Carlos Palazuelos,Javier Parcet,Mathilde Perrin

In this paper, the authors provide a combinatorial/numerical method to establish new hypercontractivity estimates in group von Neumann algebras. They illustrate their method with free groups, triangular groups and finite cyclic groups, for which they obtain optimal time hypercontractive inequalities with respect to the Markov process given by the word length and with an even integer. Interpolation and differentiation also yield general hypercontrativity for via logarithmic Sobolev inequalities. The authors' method admits further applications to other discrete groups without small loops as far as the numerical part—which varies from one group to another—is implemented and tested on a computer. The authors also develop another combinatorial method which does not rely on computational estimates and provides (non-optimal) hypercontractive inequalities for a larger class of groups/lengths, including any finitely generated group equipped with a conditionally negative word length, like infinite Coxeter groups. The authors' second method also yields hypercontractivity bounds for groups admitting a finite dimensional proper cocycle. Hypercontractivity fails for conditionally negative lengths in groups satisfying Kazhdan's property (T).

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102

Publisher

American Mathematical Soc.

Published Date

ISBN 10

1470425653

ISBN 13

9781470425654

Clifford Wavelets, Singular Integrals, and Hardy Spaces

By: Marius Mitrea

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Clifford Wavelets, Singular Integrals, and Hardy Spaces

By: Marius Mitrea

The book discusses the extensions of basic Fourier Analysis techniques to the Clifford algebra framework. Topics covered: construction of Clifford-valued wavelets, Calderon-Zygmund theory for Clifford valued singular integral operators on Lipschitz hyper-surfaces, Hardy spaces of Clifford monogenic functions on Lipschitz domains. Results are applied to potential theory and elliptic boundary value problems on non-smooth domains. The book is self-contained to a large extent and well-suited for graduate students and researchers in the areas of wavelet theory, Harmonic and Clifford Analysis. It will also interest the specialists concerned with the applications of the Clifford algebra machinery to Mathematical Physics.

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136

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Springer

Published Date

ISBN 10

3540578846

ISBN 13

9783540578840

The Hodge-Laplacian

Boundary Value Problems on Riemannian Manifolds

By: Dorina Mitrea,Irina Mitrea,Marius Mitrea,Michael Taylor

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The Hodge-Laplacian

Boundary Value Problems on Riemannian Manifolds

By: Dorina Mitrea,Irina Mitrea,Marius Mitrea,Michael Taylor

The core of this monograph is the development of tools to derive well-posedness results in very general geometric settings for elliptic differential operators. A new generation of Calderón-Zygmund theory is developed for variable coefficient singular integral operators, which turns out to be particularly versatile in dealing with boundary value problems for the Hodge-Laplacian on uniformly rectifiable subdomains of Riemannian manifolds via boundary layer methods. In addition to absolute and relative boundary conditions for differential forms, this monograph treats the Hodge-Laplacian equipped with classical Dirichlet, Neumann, Transmission, Poincaré, and Robin boundary conditions in regular Semmes-Kenig-Toro domains. Lying at the intersection of partial differential equations, harmonic analysis, and differential geometry, this text is suitable for a wide range of PhD students, researchers, and professionals. Contents: Preface Introduction and Statement of Main Results Geometric Concepts and Tools Harmonic Layer Potentials Associated with the Hodge-de Rham Formalism on UR Domains Harmonic Layer Potentials Associated with the Levi-Civita Connection on UR Domains Dirichlet and Neumann Boundary Value Problems for the Hodge-Laplacian on Regular SKT Domains Fatou Theorems and Integral Representations for the Hodge-Laplacian on Regular SKT Domains Solvability of Boundary Problems for the Hodge-Laplacian in the Hodge-de Rham Formalism Additional Results and Applications Further Tools from Differential Geometry, Harmonic Analysis, Geometric Measure Theory, Functional Analysis, Partial Differential Equations, and Clifford Analysis Bibliography Index

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671

Publisher

Walter de Gruyter GmbH & Co KG

Published Date

ISBN 10

3110483394

ISBN 13

9783110483390

Singular Integral Operators, Quantitative Flatness, and Boundary Problems

By: Juan José Marín,José María Martell,Dorina Mitrea,Irina Mitrea,Marius Mitrea

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Singular Integral Operators, Quantitative Flatness, and Boundary Problems

By: Juan José Marín,José María Martell,Dorina Mitrea,Irina Mitrea,Marius Mitrea

This monograph provides a state-of-the-art, self-contained account on the effectiveness of the method of boundary layer potentials in the study of elliptic boundary value problems with boundary data in a multitude of function spaces. Many significant new results are explored in detail, with complete proofs, emphasizing and elaborating on the link between the geometric measure-theoretic features of an underlying surface and the functional analytic properties of singular integral operators defined on it. Graduate students, researchers, and professionals interested in a modern account of the topic of singular integral operators and boundary value problems – as well as those more generally interested in harmonic analysis, PDEs, and geometric analysis – will find this text to be a valuable addition to the mathematical literature.

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605

Publisher

Springer Nature

Published Date

ISBN 10

3031082346

ISBN 13

9783031082344

The Hodge-Laplacian

Boundary Value Problems on Riemannian Manifolds

By: Dorina Mitrea,Irina Mitrea,Marius Mitrea,Michael Taylor

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The Hodge-Laplacian

Boundary Value Problems on Riemannian Manifolds

By: Dorina Mitrea,Irina Mitrea,Marius Mitrea,Michael Taylor

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671

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Walter de Gruyter GmbH & Co KG

Published Date

ISBN 10

3110483394

ISBN 13

9783110483390

Groupoid Metrization Theory

With Applications to Analysis on Quasi-Metric Spaces and Functional Analysis

By: Dorina Mitrea,Irina Mitrea,Marius Mitrea,Sylvie Monniaux

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Groupoid Metrization Theory

With Applications to Analysis on Quasi-Metric Spaces and Functional Analysis

By: Dorina Mitrea,Irina Mitrea,Marius Mitrea,Sylvie Monniaux

The topics in this research monograph are at the interface of several areas of mathematics such as harmonic analysis, functional analysis, analysis on spaces of homogeneous type, topology, and quasi-metric geometry. The presentation is self-contained with complete, detailed proofs, and a large number of examples and counterexamples are provided. Unique features of Metrization Theory for Groupoids: With Applications to Analysis on Quasi-Metric Spaces and Functional Analysis include: * treatment of metrization from a wide, interdisciplinary perspective, with accompanying applications ranging across diverse fields; * coverage of topics applicable to a variety of scientific areas within pure mathematics; * useful techniques and extensive reference material; * includes sharp results in the field of metrization. Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties. * coverage of topics applicable to a variety of scientific areas within pure mathematics; * useful techniques and extensive reference material; * includes sharp results in the field of metrization. Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties. * useful techniques and extensive reference material; * includes sharp results in the field of metrization. Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties. * includes sharp results in the field of metrization. Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties. Professional mathematicians with a wide spectrum of mathematical interests will find this book to be a useful resource and complete self-study guide. At the same time, the monograph is accessible and will be of use to advanced graduate students and to scientifically trained readers with an interest in the interplay among topology and metric properties and/or functional analysis and metric properties.

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486

Publisher

Springer Science & Business Media

Published Date

ISBN 10

0817683976

ISBN 13

9780817683979

Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds

By: Dorina Mitrea,Marius Mitrea,Michael Taylor

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Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds

By: Dorina Mitrea,Marius Mitrea,Michael Taylor

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137

Publisher

American Mathematical Soc.

Published Date

ISBN 10

082182659X

ISBN 13

9780821826591

Geometric Harmonic Analysis II

Function Spaces Measuring Size and Smoothness on Rough Sets

By: Dorina Mitrea,Irina Mitrea,Marius Mitrea

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Geometric Harmonic Analysis II

Function Spaces Measuring Size and Smoothness on Rough Sets

By: Dorina Mitrea,Irina Mitrea,Marius Mitrea

This monograph is part of a larger program, materializing in five volumes, whose principal aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings. Volume II is concerned with function spaces measuring size and/or smoothness, such as Hardy spaces, Besov spaces, Triebel-Lizorkin spaces, Sobolev spaces, Morrey spaces, Morrey-Campanato spaces, spaces of functions of Bounded Mean Oscillations, etc., in general geometric settings. Work here also highlights the close interplay between differentiability properties of functions and singular integral operators. The text is intended for researchers, graduate students, and industry professionals interested in harmonic analysis, functional analysis, geometric measure theory, and function space theory.

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938

Publisher

Springer Nature

Published Date

ISBN 10

3031137183

ISBN 13

9783031137181

Weighted Morrey Spaces

Calderón-Zygmund Theory and Boundary Problems

By: Marcus Laurel,Marius Mitrea

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Weighted Morrey Spaces

Calderón-Zygmund Theory and Boundary Problems

By: Marcus Laurel,Marius Mitrea

This monograph is a testament to the potency of the method of singular integrals of layer potential type in solving boundary value problems for weakly elliptic systems in the setting of Muckenhoupt-weighted Morrey spaces and their pre-duals. A functional analytic framework for Muckenhoupt-weighted Morrey spaces in the rough setting of Ahlfors regular sets is built from the ground up and subsequently supports a Calderón-Zygmund theory on this brand of Morrey space in the optimal geometric environment of uniformly rectifiable sets. A thorough duality theory for such Morrey spaces is also developed and ushers in a never-before-seen Calderón-Zygmund theory for Muckenhoupt-weighted Block spaces. Both weighted Morrey and Block spaces are also considered through the lens of (generalized) Banach function spaces, and ultimately, a variety of boundary value problems are formulated and solved with boundary data arbitrarily prescribed from either scale of space. The fairly self-contained nature of this monograph ensures that graduate students, researchers, and professionals in a variety of fields, e.g., function space theory, harmonic analysis, and PDE, will find this monograph a welcome and valuable addition to the mathematical literature.

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367

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Walter de Gruyter GmbH & Co KG

Published Date

ISBN 10

3111461459

ISBN 13

9783111461458

Perspectives in Partial Differential Equations, Harmonic Analysis and Applications

A Volume in Honor of Vladimir G. Maz'ya's 70th Birthday

By: Dorina Mitrea,Marius Mitrea

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Perspectives in Partial Differential Equations, Harmonic Analysis and Applications

A Volume in Honor of Vladimir G. Maz'ya's 70th Birthday

By: Dorina Mitrea,Marius Mitrea

This volume contains a collection of papers contributed on the occasion of Mazya's 70th birthday by a distinguished group of experts of international stature in the fields of harmonic analysis, partial differential equations, function theory, and spectral analysis, reflecting the state of the art in these areas.

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446

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American Mathematical Soc.

Published Date

ISBN 10

0821844245

ISBN 13

9780821844243

Multi-Layer Potentials and Boundary Problems

for Higher-Order Elliptic Systems in Lipschitz Domains

By: Irina Mitrea,Marius Mitrea

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Multi-Layer Potentials and Boundary Problems

for Higher-Order Elliptic Systems in Lipschitz Domains

By: Irina Mitrea,Marius Mitrea

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Page Count

430

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Springer

Published Date

ISBN 10

3642326668

ISBN 13

9783642326660

Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces

A Sharp Theory

By: Ryan Alvarado,Marius Mitrea

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Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces

A Sharp Theory

By: Ryan Alvarado,Marius Mitrea

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491

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Springer

Published Date

ISBN 10

3319181327

ISBN 13

9783319181325

$L^p$-Square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets

By: Steve Hofmann,Dorina Mitrea,Marius Mitrea

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$L^p$-Square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets

By: Steve Hofmann,Dorina Mitrea,Marius Mitrea

The authors establish square function estimates for integral operators on uniformly rectifiable sets by proving a local theorem and applying it to show that such estimates are stable under the so-called big pieces functor. More generally, they consider integral operators associated with Ahlfors-David regular sets of arbitrary codimension in ambient quasi-metric spaces. The local theorem is then used to establish an inductive scheme in which square function estimates on so-called big pieces of an Ahlfors-David regular set are proved to be sufficient for square function estimates to hold on the entire set. Extrapolation results for and Hardy space versions of these estimates are also established. Moreover, the authors prove square function estimates for integral operators associated with variable coefficient kernels, including the Schwartz kernels of pseudodifferential operators acting between vector bundles on subdomains with uniformly rectifiable boundaries on manifolds.

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Page Count

120

Publisher

American Mathematical Soc.

Published Date

ISBN 10

1470422603

ISBN 13

9781470422608

Book's by Marius Mitrea

Multi-Layer Potentials and Boundary Problems

Multi-Layer Potentials and Boundary Problems

Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds

Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds

Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces

Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces

Hypercontractivity in Group von Neumann Algebras

Hypercontractivity in Group von Neumann Algebras

Clifford Wavelets, Singular Integrals, and Hardy Spaces

Clifford Wavelets, Singular Integrals, and Hardy Spaces

The Hodge-Laplacian

The Hodge-Laplacian

Singular Integral Operators, Quantitative Flatness, and Boundary Problems

Singular Integral Operators, Quantitative Flatness, and Boundary Problems

The Hodge-Laplacian

The Hodge-Laplacian

Groupoid Metrization Theory

Groupoid Metrization Theory

Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds

Layer Potentials, the Hodge Laplacian, and Global Boundary Problems in Nonsmooth Riemannian Manifolds

Geometric Harmonic Analysis II

Geometric Harmonic Analysis II

Weighted Morrey Spaces

Weighted Morrey Spaces

Perspectives in Partial Differential Equations, Harmonic Analysis and Applications

Perspectives in Partial Differential Equations, Harmonic Analysis and Applications

Multi-Layer Potentials and Boundary Problems

Multi-Layer Potentials and Boundary Problems

Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces

Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces

$L^p$-Square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets

$L^p$-Square Function Estimates on Spaces of Homogeneous Type and on Uniformly Rectifiable Sets

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