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Intro; Foreword; Preface; Contents; 1 Introduction and Motivations; 2 Differential Forms; 2.1 Spaces of Functions; 2.1.1 Differential; 2.2 Differential m-Forms; 2.2.1 Operations on Differential Forms; 2.2.1.1 Sum of Differential Forms; 2.2.1.2 Exterior Product of Differential Forms; 2.2.1.3 Inner Multiplication of a Form by a Vector Field; 2.2.2 Pullback of a Form; 2.2.3 Differentiation of Forms; 2.2.3.1 Lie Derivative of a Differential Form in the Direction of a Vector Field; 2.3 Line Integrals of Differential Forms; 2.4 Surface Integrals of m-Forms; 2.4.1 Stokes Theorem
3 Currents: The Deterministic Case3.1 The Space D(U) and Its Topology; 3.2 0-Currents: Distributions; 3.3 m-Currents; 3.3.1 Operations on Currents; 3.3.1.1 Exterior Multiplication of a Current with a Vector Field; 3.3.1.2 Exterior Multiplication of a Current with a Form; 3.3.1.3 Expansion of a Current; 3.3.1.4 Cartesian Product of Currents; 3.3.2 Boundary, and Lie Derivative of a Current; 3.3.3 Push-Forward of a Current; 3.3.4 Currents Associated with Oriented Surfaces; 4 Currents: The Stochastic Case; 4.1 Random Radon Measures; 4.2 Random Radon Measures Associated with Random Closed Sets
4.2.1 Absolutely Continuous (in Mean) Random Sets4.3 Random Currents; 5 Applications; 5.1 Tumor-Driven Angiogenesis; 5.1.1 The Capillary Network; 5.1.1.1 Branching; 5.1.1.2 Anastomosis; 5.1.1.3 Mean Field Equation; 5.2 Crystal Dislocations; 5.2.1 Ensemble Averaging; 5.3 Gaussian Currents in Statistical Shape Analysis; 5.3.1 Shapes as Currents; 5.3.2 The Space of Currents on a RKHS; 5.3.2.1 The Isometric Mapping; 5.3.3 Finite Dimensional Approximation of Shapes; 5.3.4 Random Currents on Hilbert Spaces; 5.3.5 Gaussian Currents; 5.3.5.1 Statistics for Gaussian Shape Models
B.2.2 Gaussian ProcessesB.2.3 Processes with Independent Increments; B.2.4 Markov Processes; B.2.5 Brownian Motion and the Wiener Process; B.2.6 Marked Counting Processes; B.3 The Itô Integral; B.3.1 Itô Integrals of Multidimensional Wiener Processes; B.3.2 The Stochastic Differential; B.4 Multidimensional Stochastic Differentials; B.5 Stochastic Differential Equations; C Vector Calculus; C.1 m-Vectors; C.2 m-Covectors; C.2.1 Duality Pairing; C.2.2 Inner Product; C.2.3 Operations on Covectors; D Regular Surfaces; D.1 Tangent Plane, Normal Vectors, Oriented Surfaces
3 Currents: The Deterministic Case3.1 The Space D(U) and Its Topology; 3.2 0-Currents: Distributions; 3.3 m-Currents; 3.3.1 Operations on Currents; 3.3.1.1 Exterior Multiplication of a Current with a Vector Field; 3.3.1.2 Exterior Multiplication of a Current with a Form; 3.3.1.3 Expansion of a Current; 3.3.1.4 Cartesian Product of Currents; 3.3.2 Boundary, and Lie Derivative of a Current; 3.3.3 Push-Forward of a Current; 3.3.4 Currents Associated with Oriented Surfaces; 4 Currents: The Stochastic Case; 4.1 Random Radon Measures; 4.2 Random Radon Measures Associated with Random Closed Sets
4.2.1 Absolutely Continuous (in Mean) Random Sets4.3 Random Currents; 5 Applications; 5.1 Tumor-Driven Angiogenesis; 5.1.1 The Capillary Network; 5.1.1.1 Branching; 5.1.1.2 Anastomosis; 5.1.1.3 Mean Field Equation; 5.2 Crystal Dislocations; 5.2.1 Ensemble Averaging; 5.3 Gaussian Currents in Statistical Shape Analysis; 5.3.1 Shapes as Currents; 5.3.2 The Space of Currents on a RKHS; 5.3.2.1 The Isometric Mapping; 5.3.3 Finite Dimensional Approximation of Shapes; 5.3.4 Random Currents on Hilbert Spaces; 5.3.5 Gaussian Currents; 5.3.5.1 Statistics for Gaussian Shape Models
B.2.2 Gaussian ProcessesB.2.3 Processes with Independent Increments; B.2.4 Markov Processes; B.2.5 Brownian Motion and the Wiener Process; B.2.6 Marked Counting Processes; B.3 The Itô Integral; B.3.1 Itô Integrals of Multidimensional Wiener Processes; B.3.2 The Stochastic Differential; B.4 Multidimensional Stochastic Differentials; B.5 Stochastic Differential Equations; C Vector Calculus; C.1 m-Vectors; C.2 m-Covectors; C.2.1 Duality Pairing; C.2.2 Inner Product; C.2.3 Operations on Covectors; D Regular Surfaces; D.1 Tangent Plane, Normal Vectors, Oriented Surfaces