6,795 research outputs found
On the relationship between fixed points and iteration in admissible set theory without foundation
Abstract.: In this article we show how to use the result in Jäger and Probst [7] to adapt the technique of pseudo-hierarchies and its use in Avigad [1] to subsystems of set theory without foundation. We prove that the theory KPi0 of admissible sets without foundation, extended by the principle (Σ-FP), asserting the existence of fixed points of monotone Σ operators, has the same proof-theoretic ordinal as KPi0 extended by the principle (Σ-TR), that allows to iterate Σ operations along ordinals. By Jäger and Probst [6] we conclude that the metapredicative Mahlo ordinal φω00 is also the ordinal of KPi0+(Σ-FP). Hence the relationship between fixed points and iteration persists in the framework of set theory without foundatio
The visual boundary of hyperbolic free-by-cyclic groups
Let be an atoroidal outer automorphism of the free group . We
study the Gromov boundary of the hyperbolic group . We explicitly describe a family of embeddings of
the complete bipartite graph into . To do so, we
define the directional Whitehead graph and prove that an indecomposable
-tree is Levitt type if and only if one of its directional Whitehead
graphs contains more than one edge. As an application, we obtain a direct proof
of Kapovich-Kleiner's theorem that is homeomorphic to the
Menger curve if the automorphism is atoroidal and fully irreducible.Comment: 25 pages, 3 figure
A Simple Proof of Maxwell Saturation for Coupled Scalar Recursions
Low-density parity-check (LDPC) convolutional codes (or spatially-coupled
codes) were recently shown to approach capacity on the binary erasure channel
(BEC) and binary-input memoryless symmetric channels. The mechanism behind this
spectacular performance is now called threshold saturation via spatial
coupling. This new phenomenon is characterized by the belief-propagation
threshold of the spatially-coupled ensemble increasing to an intrinsic noise
threshold defined by the uncoupled system. In this paper, we present a simple
proof of threshold saturation that applies to a wide class of coupled scalar
recursions. Our approach is based on constructing potential functions for both
the coupled and uncoupled recursions. Our results actually show that the fixed
point of the coupled recursion is essentially determined by the minimum of the
uncoupled potential function and we refer to this phenomenon as Maxwell
saturation. A variety of examples are considered including the
density-evolution equations for: irregular LDPC codes on the BEC, irregular
low-density generator matrix codes on the BEC, a class of generalized LDPC
codes with BCH component codes, the joint iterative decoding of LDPC codes on
intersymbol-interference channels with erasure noise, and the compressed
sensing of random vectors with i.i.d. components.Comment: This article is an extended journal version of arXiv:1204.5703 and
has now been accepted to the IEEE Transactions on Information Theory. This
version adds additional explanation for some details and also corrects a
number of small typo
- …