Locally Collapsed 3-Manifolds

We prove that a 3-dimensional compact Riemannian manifold which is locally collapsed, with respect to a lower curvature bound, is a graph manifold. This theorem was stated by Perelman and was used in his proof of the geometrization conjecture.Comment: Final versio

On some new approaches to practical Slepian-Wolf compression inspired by channel coding

This paper considers the problem, first introduced by Ahlswede and Körner in 1975, of lossless source coding with coded side information. Specifically, let X and Y be two random variables such that X is desired losslessly at the decoder while Y serves as side information. The random variables are encoded independently, and both descriptions are used by the decoder to reconstruct X. Ahlswede and Körner describe the achievable rate region in terms of an auxiliary random variable. This paper gives a partial solution for the optimal auxiliary random variable, thereby describing part of the rate region explicitly in terms of the distribution of X and Y

Near zero modes in condensate phases of the Dirac theory on the honeycomb lattice

We investigate a number of fermionic condensate phases on the honeycomb lattice, to determine whether topological defects (vortices and edges) in these phases can support bound states with zero energy. We argue that topological zero modes bound to vortices and at edges are not only connected, but should in fact be \emph{identified}. Recently, it has been shown that the simplest s-wave superconducting state for the Dirac fermion approximation of the honeycomb lattice at precisely half filling, supports zero modes inside the cores of vortices (P. Ghaemi and F. Wilczek, 2007). We find that within the continuum Dirac theory the zero modes are not unique neither to this phase, nor to half filling. In addition, we find the \emph{exact} wavefunctions for vortex bound zero modes, as well as the complete edge state spectrum of the phases we discuss. The zero modes in all the phases we examine have even-numbered degeneracy, and as such pairs of any Majorana modes are simply equivalent to one ordinary fermion. As a result, contrary to bound state zero modes in $p_x+i p_y$ superconductors, vortices here do \emph{not} exhibit non-Abelian exchange statistics. The zero modes in the pure Dirac theory are seemingly topologically protected by the effective low energy symmetry of the theory, yet on the original honeycomb lattice model these zero modes are split, by explicit breaking of the effective low energy symmetry.Comment: Final version including numerics, accepted for publication in PR

The boundary of the outer space of a free product

Let $G$ be a countable group that splits as a free product of groups of the form $G=G_1\ast\dots\ast G_k\ast F_N$, where $F_N$ is a finitely generated free group. We identify the closure of the outer space $P\mathcal{O}(G,\{G_1,\dots,G_k\})$ for the axes topology with the space of projective minimal, \emph{very small} $(G,\{G_1,\dots,G_k\})$-trees, i.e. trees whose arc stabilizers are either trivial, or cyclic, closed under taking roots, and not conjugate into any of the $G_i$'s, and whose tripod stabilizers are trivial. Its topological dimension is equal to $3N+2k-4$, and the boundary has dimension $3N+2k-5$. We also prove that any very small $(G,\{G_1,\dots,G_k\})$-tree has at most $2N+2k-2$ orbits of branch points.Comment: v3: Final version, to appear in the Israel Journal of Mathematics. Section 3, regarding the definition and properties of geometric trees, has been rewritten to improve the exposition, following a referee's suggestio

Graph Sparsification by Edge-Connectivity and Random Spanning Trees

We present new approaches to constructing graph sparsifiers --- weighted subgraphs for which every cut has the same value as the original graph, up to a factor of $(1 \pm \epsilon)$. Our first approach independently samples each edge $uv$ with probability inversely proportional to the edge-connectivity between $u$ and $v$. The fact that this approach produces a sparsifier resolves a question posed by Bencz\'ur and Karger (2002). Concurrent work of Hariharan and Panigrahi also resolves this question. Our second approach constructs a sparsifier by forming the union of several uniformly random spanning trees. Both of our approaches produce sparsifiers with $O(n \log^2(n)/\epsilon^2)$ edges. Our proofs are based on extensions of Karger's contraction algorithm, which may be of independent interest

Generation of potential/surface density pairs in flat disks Power law distributions

We report a simple method to generate potential/surface density pairs in flat axially symmetric finite size disks. Potential/surface density pairs consist of a ``homogeneous'' pair (a closed form expression) corresponding to a uniform disk, and a ``residual'' pair. This residual component is converted into an infinite series of integrals over the radial extent of the disk. For a certain class of surface density distributions (like power laws of the radius), this series is fully analytical. The extraction of the homogeneous pair is equivalent to a convergence acceleration technique, in a matematical sense. In the case of power law distributions, the convergence rate of the residual series is shown to be cubic inside the source. As a consequence, very accurate potential values are obtained by low order truncation of the series. At zero order, relative errors on potential values do not exceed a few percent typically, and scale with the order N of truncation as 1/N**3. This method is superior to the classical multipole expansion whose very slow convergence is often critical for most practical applications.Comment: Accepted for publication in Astronomy & Astrophysics 7 pages, 8 figures, F90-code available at http://www.obs.u-bordeaux1.fr/radio/JMHure/intro2applawd.htm

Analysis of approximate nearest neighbor searching with clustered point sets

We present an empirical analysis of data structures for approximate nearest neighbor searching. We compare the well-known optimized kd-tree splitting method against two alternative splitting methods. The first, called the sliding-midpoint method, which attempts to balance the goals of producing subdivision cells of bounded aspect ratio, while not producing any empty cells. The second, called the minimum-ambiguity method is a query-based approach. In addition to the data points, it is also given a training set of query points for preprocessing. It employs a simple greedy algorithm to select the splitting plane that minimizes the average amount of ambiguity in the choice of the nearest neighbor for the training points. We provide an empirical analysis comparing these two methods against the optimized kd-tree construction for a number of synthetically generated data and query sets. We demonstrate that for clustered data and query sets, these algorithms can provide significant improvements over the standard kd-tree construction for approximate nearest neighbor searching.Comment: 20 pages, 8 figures. Presented at ALENEX '99, Baltimore, MD, Jan 15-16, 199

Correlations, inhomogeneous screening, and suppression of spin-splitting in quantum wires at strong magnetic fields

A self-consistent treatment of exchange and correlation interactions in a quantum wire (QW) subject to a strong perpendicular magnetic field is presented using a modified local-density approximation (MLDA). The influence of many-body interactions on the spin-splitting between the two lowest Landau levels (LLs) is calculated within the screened Hartree-Fock approximation (SHFA), for filling factor \nu=1, and the strong spatial dependence of the screening properties of electrons is taken into account. In comparison with the Hartree-Fock result, the spatial behavior of the occupied LL in a QW is strongly modified when correlations are included. Correlations caused by screening at the edges strongly suppress the exchange splitting and smoothen the energy dispersion at the edges. The theory accounts well for the experimentally observed strong suppression of the spin-splitting pertinent to the \nu=1 quantum Hall effect (QHE) state as well as the destruction of this state in long, quasi-ballistic GaAlAs/GaAs QWs.Comment: Text 23 pages in Latex/Revtex/preprint format, 6 Postscript figures, submitted to Physical Review