Results 1  10
of
710
A oneway quantum computer
 Phys. Rev. Lett
"... We describe a faulttolerant oneway quantum computer on cluster states in three dimensions. The presented scheme uses methods of topological error correction resulting from a link between cluster states and surface codes. The error threshold is 1.4 % for local depolarizing error and 0.11 % for each ..."
Abstract

Cited by 98 (0 self)
 Add to MetaCart
We describe a faulttolerant oneway quantum computer on cluster states in three dimensions. The presented scheme uses methods of topological error correction resulting from a link between cluster states and surface codes. The error threshold is 1.4 % for local depolarizing error and 0.11 % for each source in an error model with preparation, gate, storage and measurement errors. 1
Discrete Exterior Calculus
, 2003
"... Abstract. We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but also discrete vector fields and the operators actin ..."
Abstract

Cited by 82 (7 self)
 Add to MetaCart
Abstract. We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but also discrete vector fields and the operators acting on these objects. This allows us to address the various interactions between forms and vector fields (such as Lie derivatives) which are important in applications. Previous attempts at discrete exterior calculus have addressed only differential forms. We also introduce the notion of a circumcentric dual of a simplicial complex. The importance of dual complexes in this field has been well understood, but previous researchers have used barycentric subdivision or barycentric duals. We show that the use of circumcentric duals is crucial in arriving at a theory of discrete
On the local behavior of spaces of natural images
 Internat. J. Comput. Vision
, 2006
"... In this study we concentrate on qualitative topological analysis of the local behavior of the space of natural images. To this end, we use a space of 3 by 3 highcontrast patches M studied by Mumford et al. We develop a theoretical model for the highdensity 2dimensional submanifold of M showing th ..."
Abstract

Cited by 48 (11 self)
 Add to MetaCart
In this study we concentrate on qualitative topological analysis of the local behavior of the space of natural images. To this end, we use a space of 3 by 3 highcontrast patches M studied by Mumford et al. We develop a theoretical model for the highdensity 2dimensional submanifold of M showing that it has the topology of the Klein bottle. Using our topological software package PLEX we experimentally verify our theoretical conclusions. We use polynomial representation to give coordinatization to various subspaces of M. We find the bestfitting embedding of the Klein bottle into the ambient space of M. Our results are currently being used in developing a compression algorithm based on a Klein bottle dictionary.
Coverage and Holedetection in Sensor Networks via Homology
 Fourth International Conference on Information Processing in Sensor Networks (IPSN’05), UCLA
, 2005
"... We consider coverage problems in sensor networks of stationary nodes with minimal geometric data. In particular, there are no coordinates and no localization of nodes. We introduce a new technique for detecting holes in coverage by means of homology, an algebraic topological invariant. The impetus f ..."
Abstract

Cited by 47 (5 self)
 Add to MetaCart
We consider coverage problems in sensor networks of stationary nodes with minimal geometric data. In particular, there are no coordinates and no localization of nodes. We introduce a new technique for detecting holes in coverage by means of homology, an algebraic topological invariant. The impetus for these techniques is a completion of network communication graphs to two types of simplicial complexes: the nerve complex and the Rips complex. The former gives information about coverage intersection of individual sensor nodes, and is very difficult to compute. The latter captures connectivity in terms of internode communication: it is easy to compute but does not in itself yield coverage data. We obtain coverage data by using persistence of homology classes for Rips complexes. These homological invariants are computable: we provide simulation results. I.
Barcodes: The persistent topology of data
, 2007
"... Abstract. This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in highdimensional data. The primary mathematical tool considered is a homology theory for pointcloud data sets—persis ..."
Abstract

Cited by 43 (2 self)
 Add to MetaCart
Abstract. This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in highdimensional data. The primary mathematical tool considered is a homology theory for pointcloud data sets—persistent homology—and a novel representation of this algebraic characterization— barcodes. We sketch an application of these techniques to the classification of natural images. 1. The shape of data When a topologist is asked, “How do you visualize a fourdimensional object?” the appropriate response is a Socratic rejoinder: “How do you visualize a threedimensional object? ” We do not see in three spatial dimensions directly, but rather via sequences of planar projections integrated in a manner that is sensed if not comprehended. We spend a significant portion of our first year of life learning how to infer threedimensional spatial data from paired planar projections. Years of practice have tuned a remarkable ability to extract global structure from representations
Finding shortest nonseparating and noncontractible cycles for topologically embedded graphs
 Discrete Comput. Geom
, 2005
"... We present an algorithm for finding shortest surface nonseparating cycles in graphs embedded on surfaces in O(g 3/2 V 3/2 log V + g 5/2 V 1/2) time, where V is the number of vertices in the graph and g is the genus of the surface. If g = o(V 1/3−ε), this represents a considerable improvement over p ..."
Abstract

Cited by 43 (12 self)
 Add to MetaCart
We present an algorithm for finding shortest surface nonseparating cycles in graphs embedded on surfaces in O(g 3/2 V 3/2 log V + g 5/2 V 1/2) time, where V is the number of vertices in the graph and g is the genus of the surface. If g = o(V 1/3−ε), this represents a considerable improvement over previous results by Thomassen, and Erickson and HarPeled. We also give algorithms to find a shortest noncontractible cycle in O(g O(g) V 3/2) time, which improves previous results for fixed genus. This result can be applied for computing the (nonseparating) facewidth of embedded graphs. Using similar ideas we provide the first nearlinear running time algorithm for computing the facewidth of a graph embedded on the projective plane, and an algorithm to find the facewidth of embedded toroidal graphs in O(V 5/4 log V) time. 1
An overview of limited feedback in wireless communication systems
 IEEE J. SEL. AREAS COMMUN
, 2008
"... It is now well known that employing channel adaptive signaling in wireless communication systems can yield large improvements in almost any performance metric. Unfortunately, many kinds of channel adaptive techniques have been deemed impractical in the past because of the problem of obtaining channe ..."
Abstract

Cited by 43 (8 self)
 Add to MetaCart
It is now well known that employing channel adaptive signaling in wireless communication systems can yield large improvements in almost any performance metric. Unfortunately, many kinds of channel adaptive techniques have been deemed impractical in the past because of the problem of obtaining channel knowledge at the transmitter. The transmitter in many systems (such as those using frequency division duplexing) can not leverage techniques such as training to obtain channel state information. Over the last few years, research has repeatedly shown that allowing the receiver to send a small number of information bits about the channel conditions to the transmitter can allow near optimal channel adaptation. These practical systems, which are commonly referred to as limited or finiterate feedback systems, supply benefits nearly identical to unrealizable perfect transmitter channel knowledge systems when they are judiciously designed. In this tutorial, we provide a broad look at the field of limited feedback wireless communications. We review work in systems using various combinations of single antenna, multiple antenna, narrowband, broadband, singleuser, and multiuser technology. We also provide a synopsis of the role of limited feedback in the standardization of next generation wireless systems.
TOPOLOGICAL DEFORMATION OF HIGHER DIMENSIONAL AUTOMATA
 HOMOLOGY, HOMOTOPY AND APPLICATIONS, VOL.5(2), 2003, PP.39–82
, 2003
"... A local pospace is a gluing of topological spaces which are equipped with a closed partial ordering representing the time flow. They are used as a formalization of higher dimensional automata (see for instance [6]) which model concurrent systems in computer science. It is known [11] that there are ..."
Abstract

Cited by 41 (18 self)
 Add to MetaCart
A local pospace is a gluing of topological spaces which are equipped with a closed partial ordering representing the time flow. They are used as a formalization of higher dimensional automata (see for instance [6]) which model concurrent systems in computer science. It is known [11] that there are two distinct notions of deformation of higher dimensional automata, “spatial” and “temporal”, leaving invariant computer scientific properties like presence or absence of deadlocks. Unfortunately, the formalization of these notions is still unknown in the general case of local pospaces. We introduce here a particular kind of local pospace, the “globular CWcomplexes”, for which we formalize these notions of deformations and which are sufficient to formalize