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Calculus of variations and nonlinear partial differential equations : lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27-July 2, 2005 / / Luigi Ambrosio ; edited by Bernard Dacorogna, Paolo Marcellini
Calculus of variations and nonlinear partial differential equations : lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27-July 2, 2005 / / Luigi Ambrosio ; edited by Bernard Dacorogna, Paolo Marcellini
Autore Ambrosio Luigi
Edizione [1st ed. 2008.]
Pubbl/distr/stampa Berlin, Germany : , : Springer, , [2008]
Descrizione fisica 1 online resource (XI, 206 p.)
Disciplina 515.64
Collana C.I.M.E. Foundation Subseries
Soggetto topico Differential equations, Partial
Calculus of variations
ISBN 3-540-75914-X
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Transport Equation and Cauchy Problem for Non-Smooth Vector Fields -- Issues in Homogenization for Problems with Non Divergence Structure -- A Visit with the ?-Laplace Equation -- Weak KAM Theory and Partial Differential Equations -- Geometrical Aspects of Symmetrization -- CIME Courses on Partial Differential Equations and Calculus of Variations.
Record Nr. UNINA-9910483843703321
Ambrosio Luigi  
Berlin, Germany : , : Springer, , [2008]
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Calculus of variations and nonlinear partial differential equations : lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27-July 2, 2005 / / Luigi Ambrosio ; edited by Bernard Dacorogna, Paolo Marcellini
Calculus of variations and nonlinear partial differential equations : lectures given at the C.I.M.E. Summer School held in Cetraro, Italy, June 27-July 2, 2005 / / Luigi Ambrosio ; edited by Bernard Dacorogna, Paolo Marcellini
Autore Ambrosio Luigi
Edizione [1st ed. 2008.]
Pubbl/distr/stampa Berlin, Germany : , : Springer, , [2008]
Descrizione fisica 1 online resource (XI, 206 p.)
Disciplina 515.64
Collana C.I.M.E. Foundation Subseries
Soggetto topico Differential equations, Partial
Calculus of variations
ISBN 3-540-75914-X
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Transport Equation and Cauchy Problem for Non-Smooth Vector Fields -- Issues in Homogenization for Problems with Non Divergence Structure -- A Visit with the ?-Laplace Equation -- Weak KAM Theory and Partial Differential Equations -- Geometrical Aspects of Symmetrization -- CIME Courses on Partial Differential Equations and Calculus of Variations.
Record Nr. UNISA-996466510903316
Ambrosio Luigi  
Berlin, Germany : , : Springer, , [2008]
Materiale a stampa
Lo trovi qui: Univ. di Salerno
Opac: Controlla la disponibilità qui
Conversations on Optimal Transport
Conversations on Optimal Transport
Autore Ambrosio Luigi
Edizione [1st ed.]
Pubbl/distr/stampa Cham : , : Springer, , 2024
Descrizione fisica 1 online resource (71 pages)
Disciplina 530.138
Altri autori (Persone) QuarteroniAlfio
BonadeiFrancesca
ISBN 3-031-51685-0
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Intro -- Foreword -- From Webinars to Podcasts to Books -- Preface -- Contents -- Editors and Contributors -- About the Editors -- Contributors -- Talking about Optimal Transport -- 1 Introduction -- Reference -- Optimal Transport, Fields Medals and beyond -- 1 Introduction -- References -- From moving masses to bending spaces: an excursion in metric geometry -- 1 Introduction.
Record Nr. UNINA-9910864190503321
Ambrosio Luigi  
Cham : , : Springer, , 2024
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Gradient flows [[electronic resource] ] : in metric spaces and in the space of probability measures / / Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré
Gradient flows [[electronic resource] ] : in metric spaces and in the space of probability measures / / Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré
Autore Ambrosio Luigi
Edizione [2nd ed.]
Pubbl/distr/stampa Basel, : Birkhäuser, 2008
Descrizione fisica 1 online resource (339 p.)
Disciplina 515.42
Altri autori (Persone) GigliNicola
SavaréGiuseppe
Collana Lectures in mathematics ETH Zürich
Soggetto topico Measure theory
Metric spaces
Differential equations, Parabolic
Monotone operators
Evolution equations, Nonlinear
Soggetto genere / forma Electronic books.
ISBN 1-281-85136-1
9786611851361
3-7643-8722-X
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Notation -- Notation -- Gradient Flow in Metric Spaces -- Curves and Gradients in Metric Spaces -- Existence of Curves of Maximal Slope and their Variational Approximation -- Proofs of the Convergence Theorems -- Uniqueness, Generation of Contraction Semigroups, Error Estimates -- Gradient Flow in the Space of Probability Measures -- Preliminary Results on Measure Theory -- The Optimal Transportation Problem -- The Wasserstein Distance and its Behaviour along Geodesics -- Absolutely Continuous Curves in p(X) and the Continuity Equation -- Convex Functionals in p(X) -- Metric Slope and Subdifferential Calculus in (X) -- Gradient Flows and Curves of Maximal Slope in p(X).
Record Nr. UNINA-9910453423003321
Ambrosio Luigi  
Basel, : Birkhäuser, 2008
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Gradient flows [[electronic resource] ] : in metric spaces and in the space of probability measures / / Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré
Gradient flows [[electronic resource] ] : in metric spaces and in the space of probability measures / / Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré
Autore Ambrosio Luigi
Edizione [2nd ed.]
Pubbl/distr/stampa Basel, : Birkhäuser, 2008
Descrizione fisica 1 online resource (339 p.)
Disciplina 515.42
Altri autori (Persone) GigliNicola
SavaréGiuseppe
Collana Lectures in mathematics ETH Zürich
Soggetto topico Measure theory
Metric spaces
Differential equations, Parabolic
Monotone operators
Evolution equations, Nonlinear
ISBN 1-281-85136-1
9786611851361
3-7643-8722-X
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Notation -- Notation -- Gradient Flow in Metric Spaces -- Curves and Gradients in Metric Spaces -- Existence of Curves of Maximal Slope and their Variational Approximation -- Proofs of the Convergence Theorems -- Uniqueness, Generation of Contraction Semigroups, Error Estimates -- Gradient Flow in the Space of Probability Measures -- Preliminary Results on Measure Theory -- The Optimal Transportation Problem -- The Wasserstein Distance and its Behaviour along Geodesics -- Absolutely Continuous Curves in p(X) and the Continuity Equation -- Convex Functionals in p(X) -- Metric Slope and Subdifferential Calculus in (X) -- Gradient Flows and Curves of Maximal Slope in p(X).
Record Nr. UNINA-9910782363403321
Ambrosio Luigi  
Basel, : Birkhäuser, 2008
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Gradient flows [[electronic resource] ] : in metric spaces and in the space of probability measures / / Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré
Gradient flows [[electronic resource] ] : in metric spaces and in the space of probability measures / / Luigi Ambrosio, Nicola Gigli, Giuseppe Savaré
Autore Ambrosio Luigi
Edizione [2nd ed.]
Pubbl/distr/stampa Basel, : Birkhäuser, 2008
Descrizione fisica 1 online resource (339 p.)
Disciplina 515.42
Altri autori (Persone) GigliNicola
SavaréGiuseppe
Collana Lectures in mathematics ETH Zürich
Soggetto topico Measure theory
Metric spaces
Differential equations, Parabolic
Monotone operators
Evolution equations, Nonlinear
ISBN 1-281-85136-1
9786611851361
3-7643-8722-X
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Notation -- Notation -- Gradient Flow in Metric Spaces -- Curves and Gradients in Metric Spaces -- Existence of Curves of Maximal Slope and their Variational Approximation -- Proofs of the Convergence Theorems -- Uniqueness, Generation of Contraction Semigroups, Error Estimates -- Gradient Flow in the Space of Probability Measures -- Preliminary Results on Measure Theory -- The Optimal Transportation Problem -- The Wasserstein Distance and its Behaviour along Geodesics -- Absolutely Continuous Curves in p(X) and the Continuity Equation -- Convex Functionals in p(X) -- Metric Slope and Subdifferential Calculus in (X) -- Gradient Flows and Curves of Maximal Slope in p(X).
Record Nr. UNINA-9910823456903321
Ambrosio Luigi  
Basel, : Birkhäuser, 2008
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Lectures on Elliptic Partial Differential Equations [[electronic resource] /] / by Luigi Ambrosio, Alessandro Carlotto, Annalisa Massaccesi
Lectures on Elliptic Partial Differential Equations [[electronic resource] /] / by Luigi Ambrosio, Alessandro Carlotto, Annalisa Massaccesi
Autore Ambrosio Luigi
Edizione [1st ed. 2018.]
Pubbl/distr/stampa Pisa : , : Scuola Normale Superiore : , : Imprint : Edizioni della Normale, , 2018
Descrizione fisica 1 online resource (X, 230 p.)
Disciplina 515.353
Collana Lecture Notes (Scuola Normale Superiore)
Soggetto topico Partial differential equations
Calculus of variations
Partial Differential Equations
Calculus of Variations and Optimal Control; Optimization
ISBN 88-7642-651-5
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Variational aspects of some classes of elliptic problems -- Classical regularity theory for linear problems -- Interior regularity for nonlinear equations -- Regularity for systems -- Viscosity solutions -- Appendices - A. Some basic facts concerning Sobolev spaces -- B. Miscellaneous results in real and harmonic analysis -- C. Hausdorff measures -- D. Some tools from convex and nonsmooth analysis.
Record Nr. UNINA-9910309662103321
Ambrosio Luigi  
Pisa : , : Scuola Normale Superiore : , : Imprint : Edizioni della Normale, , 2018
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Lectures on optimal transport / / Luigi Ambrosio, Elia Brué, and Daniele Semola
Lectures on optimal transport / / Luigi Ambrosio, Elia Brué, and Daniele Semola
Autore Ambrosio Luigi
Pubbl/distr/stampa Cham, Switzerland : , : Springer, , [2021]
Descrizione fisica 1 online resource (250 pages)
Disciplina 519.6
Collana Unitext
Soggetto topico Mathematical optimization
Optimització matemàtica
Soggetto genere / forma Llibres electrònics
ISBN 3-030-72162-0
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Intro -- Preface -- Contents -- Lecture 1: Preliminary Notions and the Monge Problem -- 1 Notation and Preliminary Results -- 2 Monge's Formulation of the Optimal Transport Problem -- Lecture 2: The Kantorovich Problem -- 1 Kantorovich's Formulation of the Optimal Transport Problem -- 2 Transport Plans Versus Transport Maps -- 3 Advantages of Kantorovich's Formulation -- 4 Existence of Optimal Plans -- Lecture 3: The Kantorovich-Rubinstein Duality -- 1 Convex Analysis Tools -- 2 Proof of Duality via Fenchel-Rockafellar -- 3 The Theory of c-Duality -- 4 Proof of Duality and Dual Attainment for Bounded and Continuous Cost Functions -- Lecture 4: Necessary and Sufficient Optimality Conditions -- 1 Duality and Necessary/Sufficient Optimality Conditions for Lower Semicontinuous Costs -- 2 Remarks About Necessary and Sufficient Optimality Conditions -- 3 Remarks About c-Cyclical Monotonicity, c-Concavity and c-Transforms for Special Costs -- 4 Cost=distance2 -- 5 Cost=Distance -- 6 Convex Costs on the Real Line -- Lecture 5: Existence of Optimal Maps and Applications -- 1 Existence of Optimal Transport Maps -- 2 A Digression About Monge's Problem -- 3 Applications -- 4 Iterated Monotone Rearrangement -- Lecture 6: A Proof of the Isoperimetric Inequality and Stability in Optimal Transport -- 1 Isoperimetric Inequality -- 2 Stability of Optimal Plans and Maps -- Lecture 7: The Monge-Ampére Equation and Optimal Transport on Riemannian Manifolds -- 1 A General Change of Variables Formula -- 2 The Monge-Ampère Equation -- 3 Optimal Transport on Riemannian Manifolds -- Lecture 8: The Metric Side of Optimal Transport -- 1 The Distance W2 in P2(X) -- 2 Completeness of Square Integrable Probabilities -- 3 Characterization of Convergence in the Space of Square Integrable Probabilities.
Lecture 9: Analysis on Metric Spaces and the Dynamic Formulation of Optimal Transport -- 1 Absolutely Continuous Curves and Their Metric Derivative -- 2 Geodesics and Action -- 3 Dynamic Reformulation of the Optimal Transport Problem -- Lecture 10: Wasserstein Geodesics, Nonbranching and Curvature -- 1 Lower Semicontinuity of the Action A2 -- 2 Compactness Criterion for Curves and Random Curves -- 3 Lifting of Geodesics from X to P2(X) -- Lecture 11: Gradient Flows: An Introduction -- 1 lambda-Convex Functions -- 2 Differentiability of Absolutely Continuous Curves -- 3 Gradient Flows -- Lecture 12: Gradient Flows: The Brézis-Komura Theorem -- 1 Maximal Monotone Operators -- 2 The Implicit Euler Scheme -- 3 Reduction to Initial Conditions with Finite Energy -- 4 Discrete EVI -- Lecture 13: Examples of Gradient Flows in PDEs -- 1 p-Laplace Equation, Heat Equation in Domains, Fokker-Planck Equation -- 2 The Heat Equation in Riemannian Manifolds -- 3 Dual Sobolev Space H-1 and Heat Flow in H-1 -- Lecture 14: Gradient Flows: The EDE and EDI Formulations -- 1 EDE, EDI Solutions and Upper Gradients -- 2 Existence of EDE, EDI Solutions -- 3 Proof of Theorem 14.7 via Variational Interpolation -- Lecture 15: Semicontinuity and Convexity of Energies in the Wasserstein Space -- 1 Semicontinuity of Internal Energies -- 2 Convexity of Internal Energies -- 3 Potential Energy Functional -- 4 Interaction Energy -- 5 Functional Inequalities via Optimal Transport -- Lecture 16: The Continuity Equation and the Hopf-Lax Semigroup -- 1 Continuity Equation and Transport Equation -- 2 Continuity Equation of Geodesics in the Wasserstein Space -- 3 Hopf-Lax Semigroup -- Lecture 17: The Benamou-Brenier Formula -- 1 Benamou-Brenier Formula -- 2 Correspondence Between Absolutely Continuous Curves in the Probabilities and Solutions to the Continuity Equation.
Lecture 18: An Introduction to Otto's Calculus -- 1 Otto's Calculus -- 2 Formal Interpretation of Some Evolution Equations as Wasserstein Gradient Flows -- 3 Rigorous Interpretation of the Heat Equation as a Wasserstein Gradient Flow -- 4 More Recent Ideas and Developments -- Lecture 19: Heat Flow, Optimal Transport and Ricci Curvature -- 1 Heat Flow on Riemannian Manifolds -- 2 Heat Flow, Optimal Transport and Ricci Curvature -- References.
Record Nr. UNINA-9910495195503321
Ambrosio Luigi  
Cham, Switzerland : , : Springer, , [2021]
Materiale a stampa
Lo trovi qui: Univ. Federico II
Opac: Controlla la disponibilità qui
Lectures on optimal transport / / Luigi Ambrosio, Elia Brué, and Daniele Semola
Lectures on optimal transport / / Luigi Ambrosio, Elia Brué, and Daniele Semola
Autore Ambrosio Luigi
Pubbl/distr/stampa Cham, Switzerland : , : Springer, , [2021]
Descrizione fisica 1 online resource (250 pages)
Disciplina 519.6
Collana Unitext
Soggetto topico Mathematical optimization
Optimització matemàtica
Soggetto genere / forma Llibres electrònics
ISBN 3-030-72162-0
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Intro -- Preface -- Contents -- Lecture 1: Preliminary Notions and the Monge Problem -- 1 Notation and Preliminary Results -- 2 Monge's Formulation of the Optimal Transport Problem -- Lecture 2: The Kantorovich Problem -- 1 Kantorovich's Formulation of the Optimal Transport Problem -- 2 Transport Plans Versus Transport Maps -- 3 Advantages of Kantorovich's Formulation -- 4 Existence of Optimal Plans -- Lecture 3: The Kantorovich-Rubinstein Duality -- 1 Convex Analysis Tools -- 2 Proof of Duality via Fenchel-Rockafellar -- 3 The Theory of c-Duality -- 4 Proof of Duality and Dual Attainment for Bounded and Continuous Cost Functions -- Lecture 4: Necessary and Sufficient Optimality Conditions -- 1 Duality and Necessary/Sufficient Optimality Conditions for Lower Semicontinuous Costs -- 2 Remarks About Necessary and Sufficient Optimality Conditions -- 3 Remarks About c-Cyclical Monotonicity, c-Concavity and c-Transforms for Special Costs -- 4 Cost=distance2 -- 5 Cost=Distance -- 6 Convex Costs on the Real Line -- Lecture 5: Existence of Optimal Maps and Applications -- 1 Existence of Optimal Transport Maps -- 2 A Digression About Monge's Problem -- 3 Applications -- 4 Iterated Monotone Rearrangement -- Lecture 6: A Proof of the Isoperimetric Inequality and Stability in Optimal Transport -- 1 Isoperimetric Inequality -- 2 Stability of Optimal Plans and Maps -- Lecture 7: The Monge-Ampére Equation and Optimal Transport on Riemannian Manifolds -- 1 A General Change of Variables Formula -- 2 The Monge-Ampère Equation -- 3 Optimal Transport on Riemannian Manifolds -- Lecture 8: The Metric Side of Optimal Transport -- 1 The Distance W2 in P2(X) -- 2 Completeness of Square Integrable Probabilities -- 3 Characterization of Convergence in the Space of Square Integrable Probabilities.
Lecture 9: Analysis on Metric Spaces and the Dynamic Formulation of Optimal Transport -- 1 Absolutely Continuous Curves and Their Metric Derivative -- 2 Geodesics and Action -- 3 Dynamic Reformulation of the Optimal Transport Problem -- Lecture 10: Wasserstein Geodesics, Nonbranching and Curvature -- 1 Lower Semicontinuity of the Action A2 -- 2 Compactness Criterion for Curves and Random Curves -- 3 Lifting of Geodesics from X to P2(X) -- Lecture 11: Gradient Flows: An Introduction -- 1 lambda-Convex Functions -- 2 Differentiability of Absolutely Continuous Curves -- 3 Gradient Flows -- Lecture 12: Gradient Flows: The Brézis-Komura Theorem -- 1 Maximal Monotone Operators -- 2 The Implicit Euler Scheme -- 3 Reduction to Initial Conditions with Finite Energy -- 4 Discrete EVI -- Lecture 13: Examples of Gradient Flows in PDEs -- 1 p-Laplace Equation, Heat Equation in Domains, Fokker-Planck Equation -- 2 The Heat Equation in Riemannian Manifolds -- 3 Dual Sobolev Space H-1 and Heat Flow in H-1 -- Lecture 14: Gradient Flows: The EDE and EDI Formulations -- 1 EDE, EDI Solutions and Upper Gradients -- 2 Existence of EDE, EDI Solutions -- 3 Proof of Theorem 14.7 via Variational Interpolation -- Lecture 15: Semicontinuity and Convexity of Energies in the Wasserstein Space -- 1 Semicontinuity of Internal Energies -- 2 Convexity of Internal Energies -- 3 Potential Energy Functional -- 4 Interaction Energy -- 5 Functional Inequalities via Optimal Transport -- Lecture 16: The Continuity Equation and the Hopf-Lax Semigroup -- 1 Continuity Equation and Transport Equation -- 2 Continuity Equation of Geodesics in the Wasserstein Space -- 3 Hopf-Lax Semigroup -- Lecture 17: The Benamou-Brenier Formula -- 1 Benamou-Brenier Formula -- 2 Correspondence Between Absolutely Continuous Curves in the Probabilities and Solutions to the Continuity Equation.
Lecture 18: An Introduction to Otto's Calculus -- 1 Otto's Calculus -- 2 Formal Interpretation of Some Evolution Equations as Wasserstein Gradient Flows -- 3 Rigorous Interpretation of the Heat Equation as a Wasserstein Gradient Flow -- 4 More Recent Ideas and Developments -- Lecture 19: Heat Flow, Optimal Transport and Ricci Curvature -- 1 Heat Flow on Riemannian Manifolds -- 2 Heat Flow, Optimal Transport and Ricci Curvature -- References.
Record Nr. UNISA-996466414703316
Ambrosio Luigi  
Cham, Switzerland : , : Springer, , [2021]
Materiale a stampa
Lo trovi qui: Univ. di Salerno
Opac: Controlla la disponibilità qui
Mathematical Aspects of Evolving Interfaces [[electronic resource] ] : Lectures given at the C.I.M.-C.I.M.E. joint Euro-Summer School held in Madeira Funchal, Portugal, July 3-9, 2000 / / by Luigi Ambrosio, Klaus Deckelnick, Gerhard Dziuk, Masayasu Mimura, Vsvolod Solonnikov, Halil Mete Soner ; edited by Pierluigi Colli
Mathematical Aspects of Evolving Interfaces [[electronic resource] ] : Lectures given at the C.I.M.-C.I.M.E. joint Euro-Summer School held in Madeira Funchal, Portugal, July 3-9, 2000 / / by Luigi Ambrosio, Klaus Deckelnick, Gerhard Dziuk, Masayasu Mimura, Vsvolod Solonnikov, Halil Mete Soner ; edited by Pierluigi Colli
Autore Ambrosio Luigi
Edizione [1st ed. 2003.]
Pubbl/distr/stampa Berlin, Heidelberg : , : Springer Berlin Heidelberg : , : Imprint : Springer, , 2003
Descrizione fisica 1 online resource (XII, 248 p.)
Disciplina 515.35
Collana C.I.M.E. Foundation Subseries
Soggetto topico Partial differential equations
Differential geometry
Continuum physics
Thermodynamics
Partial Differential Equations
Differential Geometry
Classical and Continuum Physics
ISBN 3-540-39189-4
Formato Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione eng
Nota di contenuto Preface -- 1. L. Ambrosio: Lecture Notes on Optimal Transport Problems -- 2. K. Deckelnick and G. Gziuk: Numerical Approximation of Mean Curvature Flow of Graphs and Level Sets -- 3. M. Mimura: Reaction-Diffusion Systems Arising in Biological and Chemical Systems: Application of Singular Limit Procedures -- 4. V. A. Solonnikov: Lectures on Evolution Free Boundary Problems: Classical Solutions -- 5. H. M. Soner: Variational and Dynamic Problems for the Ginzburg-Landau Functional.
Record Nr. UNISA-996466501903316
Ambrosio Luigi  
Berlin, Heidelberg : , : Springer Berlin Heidelberg : , : Imprint : Springer, , 2003
Materiale a stampa
Lo trovi qui: Univ. di Salerno
Opac: Controlla la disponibilità qui