1,807 research outputs found
Analytic and Asymptotic Methods for Nonlinear Singularity Analysis: a Review and Extensions of Tests for the Painlev\'e Property
The integrability (solvability via an associated single-valued linear
problem) of a differential equation is closely related to the singularity
structure of its solutions. In particular, there is strong evidence that all
integrable equations have the Painlev\'e property, that is, all solutions are
single-valued around all movable singularities. In this expository article, we
review methods for analysing such singularity structure. In particular, we
describe well known techniques of nonlinear regular-singular-type analysis,
i.e. the Painlev\'e tests for ordinary and partial differential equations. Then
we discuss methods of obtaining sufficiency conditions for the Painlev\'e
property. Recently, extensions of \textit{irregular} singularity analysis to
nonlinear equations have been achieved. Also, new asymptotic limits of
differential equations preserving the Painlev\'e property have been found. We
discuss these also.Comment: 40 pages in LaTeX2e. To appear in the Proceedings of the CIMPA Summer
School on "Nonlinear Systems," Pondicherry, India, January 1996, (eds) B.
Grammaticos and K. Tamizhman
Balancing Minimum Spanning and Shortest Path Trees
This paper give a simple linear-time algorithm that, given a weighted
digraph, finds a spanning tree that simultaneously approximates a shortest-path
tree and a minimum spanning tree. The algorithm provides a continuous
trade-off: given the two trees and epsilon > 0, the algorithm returns a
spanning tree in which the distance between any vertex and the root of the
shortest-path tree is at most 1+epsilon times the shortest-path distance, and
yet the total weight of the tree is at most 1+2/epsilon times the weight of a
minimum spanning tree. This is the best tradeoff possible. The paper also
describes a fast parallel implementation.Comment: conference version: ACM-SIAM Symposium on Discrete Algorithms (1993
On the probabilistic min spanning tree Problem
We study a probabilistic optimization model for min spanning tree, where any vertex vi of the input-graph G(V,E) has some presence probability pi in the final instance GâČ â G that will effectively be optimized. Suppose that when this ârealâ instance GâČ becomes known, a spanning tree T, called anticipatory or a priori spanning tree, has already been computed in G and one can run a quick algorithm (quicker than one that recomputes from scratch), called modification strategy, that modifies the anticipatory tree T in order to fit G âČ. The goal is to compute an anticipatory spanning tree of G such that, its modification for any G âČ â G is optimal for G âČ. This is what we call probabilistic min spanning tree problem. In this paper we study complexity and approximation of probabilistic min spanning tree in complete graphs under two distinct modification strategies leading to different complexity results for the problem. For the first of the strategies developed, we also study two natural subproblems of probabilistic min spanning tree, namely, the probabilistic metric min spanning tree and the probabilistic min spanning tree 1,2 that deal with metric complete graphs and complete graphs with edge-weights either 1, or 2, respectively
Classification of time series by shapelet transformation
Time-series classification (TSC) problems present a specific challenge for classification algorithms: how to measure similarity between series. A \emph{shapelet} is a time-series subsequence that allows for TSC based on local, phase-independent similarity in shape. Shapelet-based classification uses the similarity between a shapelet and a series as a discriminatory feature. One benefit of the shapelet approach is that shapelets are comprehensible, and can offer insight into the problem domain. The original shapelet-based classifier embeds the shapelet-discovery algorithm in a decision tree, and uses information gain to assess the quality of candidates, finding a new shapelet at each node of the tree through an enumerative search. Subsequent research has focused mainly on techniques to speed up the search. We examine how best to use the shapelet primitive to construct classifiers. We propose a single-scan shapelet algorithm that finds the best shapelets, which are used to produce a transformed dataset, where each of the features represent the distance between a time series and a shapelet. The primary advantages over the embedded approach are that the transformed data can be used in conjunction with any classifier, and that there is no recursive search for shapelets. We demonstrate that the transformed data, in conjunction with more complex classifiers, gives greater accuracy than the embedded shapelet tree. We also evaluate three similarity measures that produce equivalent results to information gain in less time. Finally, we show that by conducting post-transform clustering of shapelets, we can enhance the interpretability of the transformed data. We conduct our experiments on 29 datasets: 17 from the UCR repository, and 12 we provide ourselve
You Can't Get Through Szekeres Wormholes - or - Regularity, Topology and Causality in Quasi-Spherical Szekeres Models
The spherically symmetric dust model of Lemaitre-Tolman can describe
wormholes, but the causal communication between the two asymptotic regions
through the neck is even less than in the vacuum
(Schwarzschild-Kruskal-Szekeres) case. We investigate the anisotropic
generalisation of the wormhole topology in the Szekeres model. The function
E(r, p, q) describes the deviation from spherical symmetry if \partial_r E \neq
0, but this requires the mass to be increasing with radius, \partial_r M > 0,
i.e. non-zero density. We investigate the geometrical relations between the
mass dipole and the locii of apparent horizon and of shell-crossings. We
present the various conditions that ensure physically reasonable
quasi-spherical models, including a regular origin, regular maxima and minima
in the spatial sections, and the absence of shell-crossings. We show that
physically reasonable values of \partial_r E \neq 0 cannot compensate for the
effects of \partial_r M > 0 in any direction, so that communication through the
neck is still worse than the vacuum.
We also show that a handle topology cannot be created by identifying
hypersufaces in the two asymptotic regions on either side of a wormhole, unless
a surface layer is allowed at the junction. This impossibility includes the
Schwarzschild-Kruskal-Szekeres case.Comment: zip file with LaTeX text + 6 figures (.eps & .ps). 47 pages. Second
replacement corrects some minor errors and typos. (First replacement prints
better on US letter size paper.
Integrable models from PT-symmetric deformations
We address the question of whether integrable models allow for PT-symmetric
deformations which preserve their intgrability. For this purpose we carry out
the Painleve test for PT-symmetric deformations of Burgers and the Korteweg-De
Vries equation. We find that the former equation allows for infinitely many
deformations which pass the Painleve test. For a specific deformation we prove
the convergence of the Painleve expansion and thus establish the Painleve
property for these models, which are therefore thought to be integrable. The
Korteweg-De Vries equation does not allow for deformations which pass the
Painleve test in complete generality, but we are able to construct a defective
Painleve expansion.Comment: 14 pages Late
Gauge Field Back-reaction on a Black Hole
The order fluctuations of gauge fields in the vicinity of a blackhole
can create a repulsive antigravity region extending out beyond the renormalized
Schwarzschild horizon. If the strength of this repulsive force increases as
higher orders in the back-reaction are included, the formation of a
wormhole-like object could occur.Comment: 17 pages, three figures available on request, in RevTe
On the equivalence between hierarchical segmentations and ultrametric watersheds
We study hierarchical segmentation in the framework of edge-weighted graphs.
We define ultrametric watersheds as topological watersheds null on the minima.
We prove that there exists a bijection between the set of ultrametric
watersheds and the set of hierarchical segmentations. We end this paper by
showing how to use the proposed framework in practice in the example of
constrained connectivity; in particular it allows to compute such a hierarchy
following a classical watershed-based morphological scheme, which provides an
efficient algorithm to compute the whole hierarchy.Comment: 19 pages, double-colum
Negaton and Positon Solutions of the KDV Equation
We give a systematic classification and a detailed discussion of the
structure, motion and scattering of the recently discovered negaton and positon
solutions of the Korteweg-de Vries equation. There are two distinct types of
negaton solutions which we label and , where is the
order of the Wronskian used in the derivation. For negatons, the number of
singularities and zeros is finite and they show very interesting time
dependence. The general motion is in the positive direction, except for
certain negatons which exhibit one oscillation around the origin. In contrast,
there is just one type of positon solution, which we label . For
positons, one gets a finite number of singularities for odd, but an
infinite number for even values of . The general motion of positons is in
the negative direction with periodic oscillations. Negatons and positons
retain their identities in a scattering process and their phase shifts are
discussed. We obtain a simple explanation of all phase shifts by generalizing
the notions of ``mass" and ``center of mass" to singular solutions. Finally, it
is shown that negaton and positon solutions of the KdV equation can be used to
obtain corresponding new solutions of the modified KdV equation.Comment: 20 pages plus 12 figures(available from authors on request),Latex
fil
On morphological hierarchical representations for image processing and spatial data clustering
Hierarchical data representations in the context of classi cation and data
clustering were put forward during the fties. Recently, hierarchical image
representations have gained renewed interest for segmentation purposes. In this
paper, we briefly survey fundamental results on hierarchical clustering and
then detail recent paradigms developed for the hierarchical representation of
images in the framework of mathematical morphology: constrained connectivity
and ultrametric watersheds. Constrained connectivity can be viewed as a way to
constrain an initial hierarchy in such a way that a set of desired constraints
are satis ed. The framework of ultrametric watersheds provides a generic scheme
for computing any hierarchical connected clustering, in particular when such a
hierarchy is constrained. The suitability of this framework for solving
practical problems is illustrated with applications in remote sensing
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