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| Mathematical Preliminaries | |
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| Vector Calculus | |
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| The Main Operations of Vector Calculus: div, grad, and � | |
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| Conservative Vector Fields | |
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| Curvilinear Integrals and the Geometric Meaning of the Existence of a Potential | |
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| Multiple and Repeated Integrals | |
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| The Flow of a Vector Field and the Gauss-Ostrogradsky Theorem | |
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| The Circulation of a Vector Field and the Green Formula | |
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| Exercises | |
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| Bibliographic Notes | |
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| Partial Differential Equations | |
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| The First Order Partial Differential Equations | |
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| The Complete Integral and the General Integral | |
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| The Singular Integral | |
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| The Quasilinear Equations and the Method of Characteristics | |
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| Compatible Systems of the First Order PDEs | |
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| The Method of Characteristics for a Non-quasilinear First Order PDE | |
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| Examples | |
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| The Second Order Partial Differential Equations | |
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| Classification of the Linear Second Order Partial Differential Equations | |
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| Boundary Value Problems for Elliptic Equations | |
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| Examples | |
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| Group Theoretic Analysis of the Systems of Partial Differential Equations | |
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| One Parameter Lie Groups | |
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| Invariance of PDEs. Systems of PDEs, and Boundary Problems under Lie Groups | |
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| Calculating a Lie Group of a PDE | |
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| Calculating Invariants of the Lie Group | |
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| Examples | |
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| Exercises | |
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| Bibliographic Notes | |
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| Theory of Generalized Convexity | |
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| Definition and Properties of the Generalized Fenchel Conjugates | |
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| Generalized Convexity and Cyclic Monotonicity | |
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| Examples | |
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| Exercises | |
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| Bibliographic Notes | |
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| Calculus of Variations and the Optimal Control | |
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| Banach Spaces and Polish Spaces | |
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| Hilbert Spaces | |
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| Dual Space for a Normed Space and a Hilbert Space | |
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| Frechet Derivative of a Mapping between Normed Spaces | |
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| Functionals and Gateaux Derivatives | |
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| Euler Equation | |
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| Optimal Control | |
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| Examples | |
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| Exercises | |
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| Bibliographic Notes | |
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| Miscellaneous Techniques | |
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| Distributions and Generalized Solutions for the Partial Differential Equations | |
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| A Motivating Example | |
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| The Set of Test Functions and its Dual | |
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| Examples of Distributions | |
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| The Derivative of a Distribution | |
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| The Product of a Distribution and a Test Function and the Product of Distributions | |
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| The Resultant of a Distribution and a Dilation Operator | |
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| Adjoint Linear Differential Operators and Generalized Solutions of the Partial Differential Equations | |
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| Sobolev Spaces and Poincare Theorem | |
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| Sweeping Operators and Balayage of Measures | |
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| Coercive Functionals | |
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| Optimization by Vector Space Methods | |
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| Calculus of Variation Problem with Convexity Constraints | |
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| Supermodularity and Monotone Comparative Statics | |
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| Hausdorff Metric on Compact Sets of a Metric Space | |
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| Generalized Envelope Theorems | |
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| Exercises | |
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| Bibliographic Notes | |
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| Economics of Multi-dimensional Screening | |
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| The Unidimensional Screening Model | |
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| Spence-Mirrlees Condition and Implementability | |
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| The Concept of the Information Rent | |
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| Three Approaches to the Unidimensional Relaxed Problem | |
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| The Direct Approach | |
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| The Dual Approach | |
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| The Hamiltonian Approach | |
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| Hamiltonian Approach to the Unidimensional Complete Problem | |
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| Type Dependent Participation Constraint | |
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| Random Participation Constraint | |
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| Examples | |
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| Exercises | |
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| Bibliographic Notes | |
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| The Multi-dimensional Screening Model | |
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| The Genericity of Exclusion | |
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| Generalized Convexity and Implementability | |
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| Is Bunching Robust in the Multi-dimensional Case? | |
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| Path Independence of Information Rents | |
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| Cost Based Tariffs | |
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| Direct Approach and Its Limitations | |
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| Dual Approach for m = n | |
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| The Relaxed Problem | |
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| An Alternative Approach to the Relaxed Problem | |
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| The Complete Problem | |
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| The Geometry of the Participation Region | |
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| A Sufficient Condition for Bunching | |
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| The Extension of the Dual Approach for n > m | |
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| Hamiltonian Approach and the First Order Characterization of the Solution | |
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| The Economic Meaning of the Lagrange Multipliers | |
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| Symmetry Analysis of the First Order Conditions | |
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| Some Remarks on the Hamiltonian Approach to the Complete Problem | |
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| Examples and Economic Applications | |
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| Exercises | |
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| Bibliographic Notes | |
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| Beyond the Quasilinear Case | |
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| The Unidimensional Case | |
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| The Multi-dimensional Case | |
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| Implementability of a Surplus Function | |
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| Implementability of an Allocation | |
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| The First Order Characterization of the Solution of the Relaxed Problem | |
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| Exercises | |
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| Bibliographic Notes | |
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| Existence, Uniqueness, and Regularity Properties of the Solution | |
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| Existence and Uniqueness of the Solution of the Relaxed Problem | |
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| Existence of a Solution for the Complete Problem | |
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| Continuity of the Solution | |
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| Bibliographic Notes | |
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| Conclusions | |