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Pure and Applied Mathematics, Volume 74: Radiative Transfer on Discrete Spaces presents the geometrical structure of natural light fields. This book describes in detail with mathematical precision the radiometric interactions of light-scattering media in terms of a few well established principles. Organized into four parts encompassing 15 chapters, this volume begins with an overview of the derivations of the practical formulas and the arrangement of formulas leading to numerical solution procedures of radiative transfer problems in plane-parallel media. This text then constructs radiative transfer theory in three ways. Other chapters consider the development of discrete radiative transfer theory from the local interaction principle. This book discusses as well the development of continuous radiative transfer theory. The final chapter deals with the task of formulating a mathematical foundation for radiative transfer theory. This book is a valuable resource for researchers in the field of radiative transfer theory whose interests transcend the physical and numerical aspects of the interaction of light with matter.

Preface Part One: Fundamentals Chapter I. Introduction 1. Radiative Transfer Theory Defined 2. Problems of Radiative Transfer Theory 3. Local and Global Formulations of the Problems 4. Continuous and Discrete Formulations of the Problems 5. Outline and Motivation for Discrete-Space Theory 6. Bibliographic Notes for Chapter I Chapter II. Geometrical Radiometry 7. Geometrical Radiometry in Radiative Transfer Theory 8. Radiant Flux 9. Geometrical Properties of Radiant Flux 10. Irradiance 11. Radiance 12. Radiance Invariants 13. Analytical Connections Among the Radiometric Concepts 14. Bibliographic Notes for Chapter II Chapter III. Radiative Transfer Theory: Continuous Formulation 15. Introduction 16. Beam Transmittance Function 17. Volume Attenuation Function 18. Volume Scattering Function 19. Path Function and Emission Function 20. Volume Absorption Function; Definition of Continuous Optical Medium in Geophysical and Astrophysical Optics 21. The Equation of Transfer 22. The Natural Solution of the Equation of Transfer 23. The General Invariant Imbedding Relation 24. The Classical Principles of Invariance 25. Functional Relations for the Operator L on General Media 26. Bibliographic Notes for Chapter III Chapter IV. The Interaction Principle 27. Introduction 28. The Interaction Principle 29. The Point-Level Interpretation 30. The Surface-Level Interpretation 31. The Space-Level Interpretation 32. The Hierarchy of Interpretations 33. The Point-Level Convention 34. Bibliographic Notes for Chapter IV Part Two: Discrete-Space Theory Chapter V. Radiative Transfer Theory: Discrete Formulation 35. Introduction 36. Special Discrete Spaces 37. General Discrete Spaces 38. Vector Formulation of the Local Interaction Principle 39. Functional Relations for the Radiance Vectors 40. Solutions of the Functional Relations 41. Scattering-Order Decomposition of the Solutions 42. Bibliographic Notes for Chapter V Chapter VI. Invariant Imbedding Relation for Discrete Spaces 43. It Will Be Shown That 44. The Divisibility Property of the Local Interaction Principle 45. Can Be Used in Hierarchies of Discrete Spaces 46. To Derive the Invariant Imbedding Relation 47. And the Principles of Invariance 48. Et Cetera 49. Bibliographic Notes for Chapter VI Part Three: Discrete-Space Applications Chapter VII. Radiative Transfer on a Linear Lattice 50. Introduction 51. The Linear Lattice 52. The Local Interaction Principle on a Linear Lattice 53. Hierarchies of Linear Lattices 54. Two-Flow Equations on a Linear Lattice 55. The Principles of Invariance on a Linear Lattice 56. Equations Governing the R and T Factors 57. Remarks on the Polarity of the R and T Factors 58. Solution of the Two-Flow Problem 59. The Plane-Parallel Medium and Its Associated Linear Lattice 60. Bibliographic Notes for Chapter VII Chapter VIII. Radiative Transfer on a Cubic Lattice 61. Introduction 62. The Extended Cubic Lattice 63. The Associated Quotient Space and Radiance Functions 64. Principles of Invariance 65. Equations Governing the R and T Operators for Multilayers 66. The R and T Operators for a Monolayer 67. Remarks on the Polarity of the R and T Operators 68. Solution of the Twenty-Six-Flow Problem 69. The Plane-Parallel Medium and Its Associated Cubic Lattice 70. Computation Procedure 71. Unification of Planetary Radiative Transfer Problems 72. Bibliographic Notes for Chapter VIII Chapter IX. Plane-Source Generated Light Fields in Discrete Spaces 73. Introduction 74. Formulation of Problem 75. The Ψ-Operator 76. First Decomposition of the Ψ-Operator 77. Complete Reflectance and Transmittance Relations 78. Second Decomposition of the Ψ-Operator 79. Details of Solution 80. Summary of Plane-Source Solution 81. Bibliographic Notes for Chapter IX Chapter X. Two Methods of Point-Source Problems in Discrete Spaces 82. Introduction 83. Formulation and Formal Solution of the Problem 84. Introduction to the Iteration Method 85. A Time-Dependent Interpretation of the Iteration Formula 86. Generalizations of the Iteration Method 87. Two Divergence Relations 88. Introduction to the Categorical Analysis Method 89. Geometry and Radiometry of Categories 90. Ψ-Operators for the Categories 91. First Decomposition of Ψ-Operators for Imbedded Categories 92. Invariant Imbedding Relation for Monoblocs 93. Principles of Invariance for Monoblocs 94. Representations of Light Field Using Complete Reflectance and Transmittance Operators on Monoblocs 95. Second Decomposition of Ψ-Operator for Monoblocs 96. Representation of Complete Operators for Monoblocs 97. Representation of the Local Ψ-Operator for Monoblocs 98. Representation of the Standard Operators for Monoblocs 99. The Categorical Analysis Concluded 100. Categorical Synthesis of the Solution 101. Bibliographic Notes for Chapter X Chapter XI. A Computer Study of Radiative Transfer on a Cubic Lattice 102. Introduction 103. The Original Physical Setting 104. The Associated Discrete Space 105. Comparison of Measured and Computed Radiances 106. Some Computer Details 107. Bibliographic Notes for Chapter XI Part Four: Advanced Topics Chapter XII. Theory of Polarized Light Fields in Discrete Spaces 108. Introduction 109. Phenomenological Definition of Polarized Radiance 110. Connections Between Standard Stokes and Standard Observable Vectors 111. Rotation Matrices 112. Scattering and Attenuation Matrices for Polarized Radiance 113. Continuous Radiative Transfer Theory for the Polarized Context 114. Discrete Radiative Transfer for the Polarized Context 115. Bibliographic Notes for Chapter XII Chapter XIII. Marcov Chains and Radiative Transfer 116. Introduction 117. Markov Chains 118. From Local Interaction Principle to Markov Chains 119. From Markov Chains to the Local Interaction Principle 120. Classification of Material-Radiative Markov Chains 121. Conclusion and Prospectus 122. Bibliographic Notes for Chapter XIII Chapter XIV. Connections with the Mainland 123. Introduction 124. The Poynting Vector and the Radiance Function 125. From Electromagnetic Theory to the Interaction Principle 126. From the Interaction Principle to the Principles of Invariance and the Equation of Transfer 127. General Connections 128. Bibliographic Notes for Chapter XIV Chapter XV. Radiative Transfer Theory: Axiomatic Formulation 129. Introduction 130. The Axioms and Their Motivations 131. Abstract Transfer Equations 132. Classical Transfer Equations 133. From the Axioms to the Interaction Principle 134. From the Axioms to the Invariant Imbedding Relation 135. From the Axioms to the Local Interaction Principle 136. Axiomatic Basis for the Theory of Polarized Radiance 137. Radiative Transfer and the Mueller Algebra 138. Summary and Prospectus 139. Bibliographic Notes for Chapter XV Chapter XVI. Some Mathematical Problems of Radiative Transfer Theory 140. Introduction 141. Statement of the Problems 142. Discussion of the Problems References Author Index Subject Index Other Titles in the Series

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