The Inverse and Ill-Posed Problems Series is a series of monographs publishing postgraduate level information on inverse and ill-posed problems for an international readership of professional scientists and researchers. The series aims to publish works which involve both theory and applications in, e.g., physics, medicine, geophysics, acoustics, electrodynamics, tomography, and ecology.
Reihe
Auflage
Sprache
Verlagsort
Verlagsgruppe
Zielgruppe
Für höhere Schule und Studium
Für Beruf und Forschung
US School Grade: College Graduate Student
Maße
Höhe: 230 mm
Breite: 155 mm
Gewicht
ISBN-13
978-90-6764-165-4 (9789067641654)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Klassifikation
INTRODUCTION
The problem of determining a metric by its hodograph and a linearization of the problem
The kinetic equation on a Riemannian manifold
Some remarks
THE RAY TRANSFORM OF SYMMETRIC TENSOR FIELDS ON EUCLIDEAN SPACE
The ray transform and its relationship to the Fourier transform
Description of the kernel of the ray transform in the smooth case
Equivalence of the first two statements of Theorem 2.2.1 in the case n = 2
Proof of Theorem 2.2.2
The ray transform of a field-distribution
Decomposition of a tensor field into potential and solenoidal parts
A theorem on the tangent component
A theorem on conjugate tensor fields on the sphere
Primality of the ideal ([x]2, (x,y))
Description of the image of the ray transform
Integral moments of the function If
Inversion formulas for the ray transform
Proof of Theorem 2.12.1
Inversion of the ray transform on the space of field-distributions
The Plancherel formula for the ray transform
Application of the ray transform to an inverse problem of photoelasticity
Further results
SOME QUESTIONS OF TENSOR ANALYSIS
Tensor fields
Covariant differentiation
Symmetric tensor fields
Semibasic tensor fields
The horizontal covariant derivative
Formulas of Gauss--Ostrogradskii type for vertical and horizontal derivatives
THE RAY TRANSFORM ON A RIEMANNIAN MANIFOLD
Compact dissipative Riemannian manifolds
The ray transform on a CDRM
The problem of inverting the ray transform
Pestov's differential identity
Poincare's inequality for semibasic tensor fields
Reduction of Theorem 4.3.3 to an inverse problem for the kinetic equation
Proof of Theorem 4.3.3
Consequences for the nonlinear problem of determining a metric from its hodograph
Bibliographical remarks
THE TRANSVERSE RAY TRANSFORM
Electromagnetic waves in quasi-isotropic media
The transverse ray transform on a CDRM
Reduction of Theorem 5.2.2 to an inverse problem for the kinetic equation
Estimation of the summand related to the right-hand side of the kinetic equation
Estimation of the boundary integral and summands depending on curvature
Proof of Theorem 5.2.2
Decomposition of the operators A0 and A1
Proof of Lemma 5.6.1
Final remarks
THE TRUNCATED TRANSVERSE RAY TRANSFORM
The polarization ellipse
The truncated transverse ray transform
Proof of Theorem 6.2.2
Decomposition of the operator Q,
Proof of Lemma 6.3.1
Inversion of the truncated transverse ray transform on Euclidean space
THE MIXED RAY TRANSFORM
Elastic waves in quasi-isotropic media
The mixed ray transform
Proof of Theorem 7.2.2
The algebraic part of the proof
THE EXPONENTIAL RAY TRANSFORM
Formulation of the main definitions and results
The modified horizontal derivative
Proof of Theorem 8.1.1
The volume of a simple compact Riemannian manifold
Determining a metric in a prescribed conformal class
Bibliographical remarks
Bibliography
Index
