Ray Transform of Symmetric Tensor Fields for a Spherically Symmetric Metric

1998 ◽  
pp. 177-187
Author(s):  
V. A. Sharafutdinov
Keyword(s):  
Symmetric Tensor ◽  
Spherically Symmetric ◽  
Tensor Fields ◽  
Ray Transform ◽  
Spherically Symmetric Metric

scholarly journals Spherically Symmetric Tensor Fields and Their Application in Nonlinear Theory of Dislocations

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 830
Author(s):  
Evgeniya V. Goloveshkina ◽  
Leonid M. Zubov
Keyword(s):  
Nonlinear Theory ◽  
Hollow Sphere ◽  
Tensor Field ◽  
Exact Analytical Solution ◽  
Symmetric Tensor ◽  
Symmetric Distribution ◽  
Spherically Symmetric ◽  
Tensor Fields ◽  
Spherically Symmetric Distribution ◽  
Nonlinear Elastic

The concept of a spherically symmetric second-rank tensor field is formulated. A general representation of such a tensor field is derived. Results related to tensor analysis of spherically symmetric fields and their geometric properties are presented. Using these results, a formulation of the spherically symmetric problem of the nonlinear theory of dislocations is given. For an isotropic nonlinear elastic material with an arbitrary spherically symmetric distribution of dislocations, this problem is reduced to a nonlinear boundary value problem for a system of ordinary differential equations. In the case of an incompressible isotropic material and a spherically symmetric distribution of screw dislocations in the radial direction, an exact analytical solution is found for the equilibrium of a hollow sphere loaded from the outside and from the inside by hydrostatic pressures. This solution is suitable for any models of an isotropic incompressible body, i. e., universal in the specified class of materials. Based on the obtained solution, numerical calculations on the effect of dislocations on the stress state of an elastic hollow sphere at large deformations are carried out.

Spherically symmetric differential equations, the rotation group, and tensor spherical functions

1969 ◽  
Vol 65 (1) ◽  
pp. 157-175 ◽  
Author(s):  
R. Burridge
Keyword(s):  
Quantum Theory ◽  
Irreducible Representation ◽  
Differential Equations ◽  
Spherical Symmetry ◽  
Symmetric Tensor ◽  
Scalar Fields ◽  
Rotation Group ◽  
Spherically Symmetric ◽  
Tensor Fields ◽  
Covariant Differentiation

AbstractIn this paper a scheme is developed for handling tensor partial differential equations having spherical symmetry. The basic technique is that of Gelfand and Shapiro ((2), §8) by which tensor fields defined on a sphere give rise to scalar fields defined on the rotation group. These fields may be expanded as series of functions, where,mis fixed and the matricesTl(g) form a 21+ 1 dimensional irreducible representation of.Spherically symmetric operations, such as covariant differentiation of tensors and the contraction of tensors with other spherically symmetric tensor fields, are shown to act in a particularly simple way on the terms of the series mentioned above: terms with givenl, nare transformed into others with the same values ofl, n. That this must be so follows from Schur's Lemma and the fact that for eachmandlthe functionsform a basis for an invariant subspace of functions onof dimension 2l+ 1 in which an irreducible representation ofacts. Explicit formulae for the results of such operations are presented.The results are used to show the existence of scalar potentials for tensors of all ranks and the results for tensors of the second rank are shown to be closely related to those recently obtained by Backus(1).This work is intended for application in geophysics and other fields where spherical symmetry plays an important role. Since workers in these fields may not be familiar with quantum theory, some matter in sections 2–5 has been included in spite of the fact that it is well known in the quantum theory of angular momentum.

scholarly journals Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Venkateswaran P. Krishnan ◽  
Vladimir A. Sharafutdinov
Keyword(s):  
Fourier Transform ◽  
Sobolev Spaces ◽  
Tensor Field ◽  
Differential Operators ◽  
Symmetric Tensor ◽  
Higher Order ◽  
Tensor Fields ◽  
Ray Transform ◽  
The Fourier Transform ◽  
Derivatives Of

<p style='text-indent:20px;'>For an integer <inline-formula><tex-math id="M1">\begin{document}$ r\ge0 $\end{document}</tex-math></inline-formula>, we prove the <inline-formula><tex-math id="M2">\begin{document}$ r^{\mathrm{th}} $\end{document}</tex-math></inline-formula> order Reshetnyak formula for the ray transform of rank <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula> symmetric tensor fields on <inline-formula><tex-math id="M4">\begin{document}$ {{\mathbb R}}^n $\end{document}</tex-math></inline-formula>. Roughly speaking, for a tensor field <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula>, the order <inline-formula><tex-math id="M6">\begin{document}$ r $\end{document}</tex-math></inline-formula> refers to <inline-formula><tex-math id="M7">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-integrability of higher order derivatives of the Fourier transform <inline-formula><tex-math id="M8">\begin{document}$ \widehat f $\end{document}</tex-math></inline-formula> over spheres centered at the origin. Certain differential operators <inline-formula><tex-math id="M9">\begin{document}$ A^{(m,r,l)}\ (0\le l\le r) $\end{document}</tex-math></inline-formula> on the sphere <inline-formula><tex-math id="M10">\begin{document}$ {{\mathbb S}}^{n-1} $\end{document}</tex-math></inline-formula> are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any <inline-formula><tex-math id="M11">\begin{document}$ r $\end{document}</tex-math></inline-formula> although the volume of calculations grows fast with <inline-formula><tex-math id="M12">\begin{document}$ r $\end{document}</tex-math></inline-formula>. The algorithm is realized for small values of <inline-formula><tex-math id="M13">\begin{document}$ r $\end{document}</tex-math></inline-formula> and Reshetnyak formulas of orders <inline-formula><tex-math id="M14">\begin{document}$ 0,1,2 $\end{document}</tex-math></inline-formula> are presented in an explicit form.</p>

scholarly journals Microlocal inversion of a 3-dimensional restricted transverse ray transform on symmetric tensor fields

2021 ◽  
Vol 495 (1) ◽  
pp. 124700
Author(s):  
Venkateswaran P. Krishnan ◽  
Rohit Kumar Mishra ◽  
Suman Kumar Sahoo
Keyword(s):  
Symmetric Tensor ◽  
Tensor Fields ◽  
Ray Transform ◽  
3 Dimensional

scholarly journals A support theorem for the geodesic ray transform of symmetric tensor fields

2009 ◽  
Vol 3 (3) ◽  
pp. 453-464 ◽  
Author(s):  
Venkateswaran P. Krishnan ◽  
◽  
Plamen Stefanov ◽  
Keyword(s):  
Symmetric Tensor ◽  
Tensor Fields ◽  
Support Theorem ◽  
Ray Transform ◽  
Geodesic Ray Transform

scholarly journals A Note on the Transformability of Spherically Symmetric Metrics

1953 ◽  
Vol 9 (1) ◽  
pp. 13-16 ◽  
Author(s):  
Paul Kustaanheimo
Keyword(s):  
Spherically Symmetric ◽  
Symmetric Metrics ◽  
Spherically Symmetric Metric

SummaryIt is shown that every spherically symmetric metric can be transformed into the isotropic form. As illustration an example is given.

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