scholarly journals The ray transform of symmetric tensor fields with incomplete projection data, I: The kernel of the ray transform

2021 ◽  
Vol 18 (2) ◽  
pp. 1219-1237
Author(s):  
V. A. Sharafutdinov
Keyword(s):  
Symmetric Tensor ◽  
Projection Data ◽  
Tensor Fields ◽  
Ray Transform

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 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.

The topology of symmetric tensor fields

1997 ◽  
Author(s):  
Lambertus Hesselink ◽  
Yingmei Lavin ◽  
Rajesh Batra ◽  
Yuval Levy ◽  
Lambertus Hesselink ◽  
...  
Keyword(s):  
Symmetric Tensor ◽  
Tensor Fields

UNCONSTRAINED HIGHER SPINS IN FOUR DIMENSIONS

2011 ◽  
Vol 08 (03) ◽  
pp. 511-556 ◽  
Author(s):  
GIUSEPPE BANDELLONI
Keyword(s):  
Equations Of Motion ◽  
Symmetric Tensor ◽  
Tensor Fields ◽  
Geometrical Meaning ◽  
Four Dimensions ◽  
Spin Field ◽  
Angular Momenta ◽  
Higher Spin ◽  
Spin Fields ◽  
The Right

The relativistic symmetric tensor fields are, in four dimensions, the right candidates to describe Higher Spin Fields. Their highest spin content is isolated with the aid of covariant conditions, discussed within a group theory framework, in which auxiliary fields remove the lower intrinsic angular momenta sectors. These conditions are embedded within a Lagrangian Quantum Field theory which describes an Higher Spin Field interacting with a Classical background. The model is invariant under a (B.R.S.) symmetric unconstrained tensor extension of the reparametrization symmetry, which include the Fang–Fronsdal algebra in a well defined limit. However, the symmetry setting reveals that the compensator field, which restore the Fang–Fronsdal symmetry of the free equations of motion, is in the existing in the framework and has a relevant geometrical meaning. The Ward identities coming from this symmetry are discussed. Our constraints give the result that the space of the invariant observables is restricted to the ones constructed with the Highest Spin Field content. The quantum extension of the symmetry reveals that no new anomaly is present. The role of the compensator field in this result is fundamental.

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