476 research outputs found
Conjugate Points in Length Spaces
In this paper we extend the concept of a conjugate point in a Riemannian
manifold to complete length spaces (also known as geodesic spaces). In
particular, we introduce symmetric conjugate points and ultimate conjugate
points. We then generalize the long homotopy lemma of Klingenberg to this
setting as well as the injectivity radius estimate also due to Klingenberg
which was used to produce closed geodesics or conjugate points on Riemannian
manifolds. Our versions apply in this more general setting. We next focus on
spaces, proving Rauch-type comparison theorems. In
particular, much like the Riemannian setting, we prove an Alexander-Bishop
theorem stating that there are no ultimate conjugate points less than
apart in a space. We also prove a relative Rauch comparison
theorem to precisely estimate the distance between nearby geodesics. We close
with applications and open problems.Comment: 47 pages, 10 figures, added references and comments to prior notion
Conjugate Points in Length Spaces
In this paper we extend the concept of a conjugate point in a Riemannian
manifold to complete length spaces (also known as geodesic spaces). In
particular, we introduce symmetric conjugate points and ultimate conjugate
points. We then generalize the long homotopy lemma of Klingenberg to this
setting as well as the injectivity radius estimate also due to Klingenberg
which was used to produce closed geodesics or conjugate points on Riemannian
manifolds. Our versions apply in this more general setting. We next focus on
spaces, proving Rauch-type comparison theorems. In
particular, much like the Riemannian setting, we prove an Alexander-Bishop
theorem stating that there are no ultimate conjugate points less than
apart in a space. We also prove a relative Rauch comparison
theorem to precisely estimate the distance between nearby geodesics. We close
with applications and open problems.Comment: 47 pages, 10 figures, added references and comments to prior notion
Approximate Linear Time ML Decoding on Tail-Biting Trellises in Two Rounds
A linear time approximate maximum likelihood decoding algorithm on
tail-biting trellises is prsented, that requires exactly two rounds on the
trellis. This is an adaptation of an algorithm proposed earlier with the
advantage that it reduces the time complexity from O(mlogm) to O(m) where m is
the number of nodes in the tail-biting trellis. A necessary condition for the
output of the algorithm to differ from the output of the ideal ML decoder is
reduced and simulation results on an AWGN channel using tail-biting rrellises
for two rate 1/2 convoluational codes with memory 4 and 6 respectively are
reporte
- …