Problem Of Integral Geometry Research Articles

Integral geometry problem for nontrapping manifolds

We consider the integral geometry problem of restoring a tensor field on a manifold with boundary from its integrals over geodesics running between boundary points. For nontrapping manifolds with a certain upper curvature bound, we prove that a tensor field, integrating to zero over geodesics between boundary points, is potential. For functions and 1-forms, this is shown to be true for arbitrary nontrapping manifolds with no conjugate points. As a consequence, we also establish deformation boundary rigidity for strongly geodesic minimizing manifolds with a certain upper curvature bound.

Riesz potentials and integral geometry in the space of rectangular matrices

Riesz potentials on the space of rectangular n × m matrices arise in diverse “higher rank” problems of harmonic analysis, representation theory, and integral geometry. In the rank-one case m = 1 they coincide with the classical operators of Marcel Riesz. We develop new tools and obtain a number of new results for Riesz potentials of functions of matrix argument. The main topics are the Fourier transform technique, representation of Riesz potentials by convolutions with a positive measure supported by submanifolds of matrices of rank < m , the behavior on smooth and L p functions. The results are applied to investigation of Radon transforms on the space of real rectangular matrices.

Dynamical approach to some problems in integral geometry

As is well known, the main problem in integral geometry is to reconstruct a function in a given domain D D , where its integrals over a family of subdomains in D D are known. Such a problem is interesting not only as an object of pure analysis, but also in connection with various applications in practical disciplines. The most remarkable example of such a connection is the Radon problem and tomography. In this paper we solve one of these problems when D D is a bounded domain in R 2 {\mathbb {R}}^2 with a piecewise smooth boundary. Some intermediate results related to dynamical systems with two generators and to some functional-integral equations are new and interesting per se. As an application of the results obtained we briefly study a boundary problem for a general third order hyperbolic partial differential equation in a bounded domain D ⊂ R 2 D\subset {\mathbb {R}}^2 with data on the whole boundary ∂ D \partial D .

The circle transform on trees

We consider the overdetermined problem of integral geometry on trees given by the transform that integrates functions on a tree over circles, and exhibit difference equations that describe the range. We then show how this problem modifies if we restrict the transform to some natural subcomplex of the complex of circles, proving inversion formulas and characterizing ranges.

Some problems of integral geometry on Anosov manifolds

In this paper we prove that on an Anosov manifold the space of symmetric m-tensor fields of vanishing energy is finite dimensional modulo the space of potential tensor fields for an arbitrary m and coincides with the latter for m=0 and m=1. For m=2 this question relates to the spectral rigidity problem.

Formulas for the inverse problem for kinetic equation and for integral geometry

Abstract - In this paper we consider the formulas for solving the problem of integral geometry in which the desired function is integrated along the geodesics of the Riemann metric, which connect points of a fixed straight line.

Analog of the F. and M. Riesz theorem for A-analytic functions

Abstract - The analog of the F. and M. Riesz theorem is proved for functions satisfying the operator equation of the Beltrami type. Such functions are called A-analytic. The interest to this class of functions is stimulated by the fact that they can be used for solving a wide class of boundary-value problems for second order elliptic systems of equations and integral geometry problems on the plane

On symmetry condition of images of Fourier – Gelfand – Graev integral transformation

Abstract - Considering some integral geometry problems it is used a so-called Fourier - Gelfand - Graev transformation. This mapping transforms an arbitrary function defined on the Lobachevsky space into a homogeneous function on a cone. For a function to be the Fourier -Gelfand -Graev image it is necessary and sufficient for some condition (symmetry condition) to be valid. In the present paper different forms of the symmetry condition are studied. A relation between the symmetry condition and a theorem of the expansion in continuous spectrum eigenfunctions of the Beltrami - Laplace operator on the Lobachevsky space is established.

Integral geometry problems for symmetric tensor fields with incomplete data

Abstract - The paper is devoted to the reconstruction problems of a compactly supported symmetric tensor field in ℝn , if its ray transform is known along the lines, intersecting a given curve. Two versions of the reconstruction are considered.

Stability of a solution to an integral geometry problem in Sobolev norms

Stability of a solution to an integral geometry problem in Sobolev norms

Some problems of the theory of multidimensional inverse problems

Abstract - In this paper, some methods of solution of multidimensional inverse problems are developed. The results of constructive solutions of inverse problems are presented. In particular, inverse problems for differential equations of the second order are considered. Furthermore, problems of integral geometry and problems of determination of convex surfaces are considered. In the last case, the functionals of them are given in the plane. This paper also concerns solution of some other inverse problems.

A Problem of Integral Geometry Related to a Triple of Grassmann Manifolds

A Problem of Integral Geometry Related to a Triple of Grassmann Manifolds

A perturbed integral geometry problem in three-dimensional space

p(x) is the projection of the paraboloid ~(x) onto the plane xa = 0; and P(x) is the part of the layer S which is bounded by the surface of ~(x) and the plane x3 = 0. Consider the operator equation in a function u:

Integral Geometry on Grassmann Manifolds and Calculus of Invariant Differential Operators

In this paper, we mainly deal with two problems in integral geometry, the range characterizations and construction of inversion formulas for Radon transforms on higher rank Grassmann manifolds. The results will be described explicitly in terms of invariant differential operators. We will also study the harmonic analysis on Grassmann manifolds, using the method of integral geometry. In particular, we will give eigenvalue formulas and radial part formulas for invariant differential operators.

An application of radon inversion to diffraction tomography

Basic equations of linear computer tomography are considered; a connection of the mathermatical formulation of problems of computer tomography with problems of integral geometry is indicated; the properties of the Radon transform for even- and odd-dimensional spaces are analyzed, and applications of Radon inversion to problems of diffraction tomography are presented. Bibliography: 5 titles.

Operator Volterra equations and integral geometry problems

Operator Volterra equations and integral geometry problems

Volterra-type integral geometry problems on the plane for curves with singularities

Volterra-type integral geometry problems on the plane for curves with singularities

Integral geometry of a tensor field on a surface of revolution

from the known integrals of f over the geodesics of metric (0.1) which join the points of the circle ρ = ρ1. His method, exposed in detail in § 3.4 of the book [4], also bases on the above-mentioned geometric fact. Due to this fact, some Volterra integral equation of second kind appears for each Fourier coefficient of the function f(ρ cos θ, ρ sin θ). This equation implies uniqueness of a solution to the problem under assumption (0.2). It was shown in § 3.7 of [4] that the linearization of the inverse kinematic problem for anisotropic media leads to the integral geometry problem for a symmetric tensor field of second degree. As was mentioned there, a solution to the latter is not unique. Theorem 3.6 of [4] states only that some part of components of a sought tensor field can be recovered under the assumption that the other components are known. The integral geometry problem for tensor fields of higher degree is of interest in its own right. For instance, it was shown in Chapter 7 of [5] that, in the case of quasi-isotropic elastic media, the problem of determining the anisotropic part of the elasticity tensor from the results of measuring the phase of a compression wave is equivalent to the integral geometry problem for a symmetric tensor field of fourth degree. The present article treats the integral geometry problem of symmetric tensor fields along the geodesics of a spherically symmetric metric. We completely describe the extent of nonuniqueness for a solution to the problem. The main result states

Integral geometry problems with perturbation on the plane

Integral geometry problems with perturbation on the plane

The integral geometry problem for a family of cones in then-dimensional space

The integral geometry problem for a family of cones in then-dimensional space