Search results for "Injective function"
showing 10 items of 18 documents
The proof of Birman’s conjecture on singular braid monoids
2003
Let B_n be the Artin braid group on n strings with standard generators sigma_1, ..., sigma_{n-1}, and let SB_n be the singular braid monoid with generators sigma_1^{+-1}, ..., sigma_{n-1}^{+-1}, tau_1, ..., tau_{n-1}. The desingularization map is the multiplicative homomorphism eta: SB_n --> Z[B_n] defined by eta(sigma_i^{+-1}) =_i^{+-1} and eta(tau_i) = sigma_i - sigma_i^{-1}, for 1 <= i <= n-1. The purpose of the present paper is to prove Birman's conjecture, namely, that the desingularization map eta is injective.
Abel transforms with low regularity with applications to X-ray tomography on spherically symmetric manifolds
2017
We study ray transforms on spherically symmetric manifolds with a piecewise $C^{1,1}$ metric. Assuming the Herglotz condition, the X-ray transform is injective on the space of $L^2$ functions on such manifolds. We also prove injectivity results for broken ray transforms (with and without periodicity) on such manifolds with a $C^{1,1}$ metric. To make these problems tractable in low regularity, we introduce and study a class of generalized Abel transforms and study their properties. This low regularity setting is relevant for geophysical applications.
Remarks on (Q, P, Y)-Summing Operators
2003
Abstract unavailable at this time... Mathematics Subject Classification (1991): 47B10. Key words: Summing operators; injective tensor product. Quaestiones Mathematicae 26(2003), 97-103
Generalized Braid Groups and Mapping Class Gropus
1997
Given a chord system of D2, we associate a generalized braid group, a surface and a homomorphism from this braid group to the mapping class group of the surface. We disprove a conjecture stated in an article by Perron and Vannier by showing that generally this homomorphism is not injective.
The complex of words and Nakaoka stability
2005
We give a new simple proof of the exactness of the complex of injective words and use it to prove Nakaoka's homology stability for symmetric groups. The methods are generalized to show acyclicity in low degrees for the complex of words in "general position". Hm(§ni1;Z) = Hm(§n;Z) for n=2 > m where §n denotes the permutation group of n elements. An elementary proof of this fact has not been available in the literature. In the first section the complex C⁄(m) of abelian groups is studied which in de- gree n is freely generated by injective words of length n. The alphabet consists of m letters. The complex C⁄(m) has the only non vanishing homology in degree m (Theorem 1). This is a result of F.…
On constructing injective spaces of type C(K)
1998
Abstract In this paper we give a general method to construct averaging operators from which we obtain almost all known methods to obtain injective spaces of type C(K). From this point of view, some known constructions are better understood and they can be easily generalized and simplified, and we also obtain some new examples of injective spaces that have not been considered before.
Coherent Quantum Tomography
2016
We discuss a quantum mechanical indirect measurement method to recover a position dependent Hamilton matrix from time evolution of coherent quantum mechanical states through an object. A mathematical formulation of this inverse problem leads to weighted X-ray transforms where the weight is a matrix. We show that such X-ray transforms are injective with very rough weights. Consequently, we can solve our quantum mechanical inverse problem in several settings, but many physically relevant problems we pose also remain open. We discuss the physical background of the proposed imaging method in detail. We give a rigorous mathematical treatment of a neutrino tomography method that has been previous…
Tensor tomography in periodic slabs
2018
Abstract The X-ray transform on the periodic slab [ 0 , 1 ] × T n , n ≥ 0 , has a non-trivial kernel due to the symmetry of the manifold and presence of trapped geodesics. For tensor fields gauge freedom increases the kernel further, and the X-ray transform is not solenoidally injective unless n = 0 . We characterize the kernel of the geodesic X-ray transform for L 2 -regular m -tensors for any m ≥ 0 . The characterization extends to more general manifolds, twisted slabs, including the Mobius strip as the simplest example.
On Radon transforms on compact Lie groups
2016
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 $S^1$ nor to $S^3$. 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 $S^1$.
A reflection approach to the broken ray transform
2013
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 $S^n$. 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 t…