61 research outputs found
Boundary reconstruction for the broken ray transform
We reduce boundary determination of an unknown function and its normal
derivatives from the (possibly weighted and attenuated) broken ray data to the
injectivity of certain geodesic ray transforms on the boundary. For
determination of the values of the function itself we obtain the usual geodesic
ray transform, but for derivatives this transform has to be weighted by powers
of the second fundamental form. The problem studied here is related to
Calder\'on's problem with partial data.Comment: 23 pages, 1 figure; final versio
On Radon transforms on finite groups
If is a finite group, is a function determined by its
sums over all cosets of cyclic subgroups of ? In other words, is the Radon
transform on injective? This inverse problem is a discrete analogue of
asking whether a function on a compact Lie group is determined by its integrals
over all geodesics. We discuss what makes this new discrete inverse problem
analogous to well-studied inverse problems on manifolds and we also present
some alternative definitions. We use representation theory to prove that the
Radon transform fails to be injective precisely on Frobenius complements. We
also give easy-to-check sufficient conditions for injectivity and
noninjectivity for the Radon transform, including a complete answer for abelian
groups and several examples for nonabelian ones.Comment: 23 page
On Radon transforms on tori
We show injectivity of the X-ray transform and the -plane Radon transform
for distributions on the -torus, lowering the regularity assumption in the
recent work by Abouelaz and Rouvi\`ere. We also show solenoidal injectivity of
the X-ray transform on the -torus for tensor fields of any order, allowing
the tensors to have distribution valued coefficients. These imply new
injectivity results for the periodic broken ray transform on cubes of any
dimension.Comment: 13 page
A reflection approach to the broken ray transform
We reduce the broken ray transform on some Riemannian manifolds (with
corners) to the geodesic ray transform on another manifold, which is obtained
from the original one by reflection. We give examples of this idea and present
injectivity results for the broken ray transform using corresponding earlier
results for the geodesic ray transform. Examples of manifolds where the broken
ray transform is injective include Euclidean cones and parts of the spheres
. In addition, we introduce the periodic broken ray transform and use the
reflection argument to produce examples of manifolds where it is injective. We
also give counterexamples to both periodic and nonperiodic cases. The broken
ray transform arises in Calder\'on's problem with partial data, and we give
implications of our results for this application.Comment: 29 pages, 6 figures; final versio
Broken ray transform on a Riemann surface with a convex obstacle
We consider the broken ray transform on Riemann surfaces in the presence of
an obstacle, following earlier work of Mukhometov. If the surface has
nonpositive curvature and the obstacle is strictly convex, we show that a
function is determined by its integrals over broken geodesic rays that reflect
on the boundary of the obstacle. Our proof is based on a Pestov identity with
boundary terms, and it involves Jacobi fields on broken rays. We also discuss
applications of the broken ray transform.Comment: 24 pages, 2 figure
Urysohn's metrization theorem for higher cardinals
In this paper a generalization of Urysohn's metrization theorem is given for
higher cardinals. Namely, it is shown that a topological space with a basis of
cardinality at most or smaller is -metrizable if and
only if it is -additive and regular, or, equivalently,
-additive, zero-dimensional, and T\textsubscript{0}. Furthermore,
all such spaces are shown to be embeddable in a suitable generalization of
Hilbert's cube.Comment: 8 pages, no figure
On Radon transforms on compact Lie groups
We show that the Radon transform related to closed geodesics is injective on
a Lie group if and only if the connected components are not homeomorphic to
nor to . This is true for both smooth functions and distributions.
The key ingredients of the proof are finding totally geodesic tori and
realizing the Radon transform as a family of symmetric operators indexed by
nontrivial homomorphisms from .Comment: 13 page
- …