相关论文: A uniqueness result for the inverse problem of ide…
We formulate indefinite integration with respect to an irregular function as an algebraic problem and provide a criterion for the existence and uniqueness of a solution. This allows us to define a good notion of integral with respect to…
Integral geometry deals with those integral transforms which associate to ``functions'' on a manifold their integrals along submanifolds parameterized by another manifold. Basic problems in this context are range characterization--where…
Let $\mR$ be the restriction of the spherical Radon transform to the set of spheres centered on a hypersurface $\mS$. We study the inversion of $\mR$ by a closed-form formula. We approach the problem by studying an oscillatory integral,…
An inverse problem of elasticity of $n$ elastic inclusions embedded into an elastic half-plane is analyzed. The boundary of the half-plane is free of traction. The half-plane and the inclusions are subjected to antiplane shear, and the…
In this article, we characterize the strength of the reconstructed singularities and artifacts in a reconstruction formula for limited data spherical Radon transform. Namely, we assume that the data is only available on a closed subset…
The traditional approaches to computerized tomography (CT) depend on the samples of Radon transform at multiple angles. In optics, the real time imaging requires the reconstruction of an object by the samples of Radon transform at a single…
In this paper, we consider the problem of finding the hypersurface M^n in the Euclidean (n+1)-space R^{n+1} that satisfies an equation of mean curvature type, called singular minimal hypersurface equation. Such an equation physically…
We present novel microlocal and injectivity analyses of ellipsoid and hyperboloid Radon transforms. We introduce a new Radon transform, $R$, which defines the integrals of a compactly supported $L^2$ function, $f$, over ellipsoids and…
We give a definition of Radon hypergeometric function (Radon HGF) of confluent and nonconfluent type, which is a function on the Grassmannian Gr(m,nr) obtained as a Radon transform of a character of the universal covering group of…
Our aim is to study invariant hypersurfaces immersed in the Euclidean space $\mathbb{R}^{n+1}$, whose mean curvature is given as a linear function in the unit sphere $\mathbb{S}^n$ depending on its Gauss map. These hypersurfaces are closely…
Certain topics on polygons are extended from Euclidean to hyperbolic geometry. This first part deals with uniqueness and existence of cocyclic polygons with prescribed sidelengths. The non-Euclidean versions are more difficult due to the…
Several years ago the authors started looking at some problems of convex geometry from a more general point of view, replacing volume by an arbitrary measure. This approach led to new general properties of the Radon transform on convex…
Vector calculus in three dimensions with a Euclidian metric is the lingua franca of classical physics, including classical electrodynamics. This article corrects some long-standing imprecision in a fundamental result. Some textbooks assert…
The isometric embedding problem is a fundamental problem in differential geometry. A longstanding problem is considered in this paper to characterize intrinsic metrics on a two-dimensional Riemannian manifold which can be realized as…
The goal of this paper is to describe the $\alpha$-cosine transform on functions on a Grassmannian of $i$-planes in an $n$-dimensional real vector space. in analytic terms as explicitly as possible. We show that for all but finitely many…
Classical integral geometry takes place in Euclidean space, but one can attempt to imitate it in any other metric space. In particular, one can attempt this in R^n equipped with the metric derived from the p-norm. This has, in effect, been…
We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs. The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics. The main tool…
The article suggests a new approach what is called a consistency method for the inversion of the spherical Radon transform in 2D with detectors on a line. It is known that there is not an exact inversion formula in 2D. By means of the…
We study an inverse uniqueness with a knowledge of spectral data in the interior transmission problem defined by an index of refraction in a simple domain. We expand the solution in such a domain into a series of one dimensional problems.…
We consider integrals of spherical harmonics with Fourier exponents on the sphere $S^n ,\, n \geq 1$. Such transforms arise in the framework of the theory of weighted Radon transforms and vector diffraction in electromagnetic fields theory.…