244,103 research outputs found
Why Use Sobolev Metrics on the Space of Curves
We study reparametrization invariant Sobolev metrics on spaces of regular curves. We discuss their completeness properties and the resulting usability for applications in shape analysis. In particular, we will argue, that the development of efficient numerical methods for higher order Sobolev type metrics is an extremely desirable goal
Killing spectroscopy of closed timelike curves
We analyse the existence of closed timelike curves in spacetimes which
possess an isometry. In particular we check which discrete quotients of such
spaces lead to closed timelike curves. As a by-product of our analysis, we
prove that the notion of existence or non-existence of closed timelike curves
is a T-duality invariant notion, whenever the direction along which we apply
such transformations is everywhere spacelike. Our formalism is
straightforwardly applied to supersymmetric theories. We provide some new
examples in the context of D-branes and generalized pp-waves.Comment: 1+35 pages, no figures; v2, new references added. Final version to
appear in JHE
Symetric Monopoles
We discuss Bogomolny monopoles of arbitrary charge invariant
under various symmetry groups. The analysis is largely in terms of the spectral
curves, the rational maps, and the Nahm equations associated with monopoles. We
consider monopoles invariant under inversion in a plane, monopoles with cyclic
symmetry, and monopoles having the symmetry of a regular solid. We introduce
the notion of a strongly centred monopole and show that the space of such
monopoles is a geodesic submanifold of the monopole moduli space.
By solving Nahm's equations we prove the existence of a tetrahedrally
symmetric monopole of charge and an octahedrally symmetric monopole of
charge , and determine their spectral curves. Using the geodesic
approximation to analyse the scattering of monopoles with cyclic symmetry, we
discover a novel type of non-planar -monopole scattering process
Conformally invariant scaling limits in planar critical percolation
This is an introductory account of the emergence of conformal invariance in
the scaling limit of planar critical percolation. We give an exposition of
Smirnov's theorem (2001) on the conformal invariance of crossing probabilities
in site percolation on the triangular lattice. We also give an introductory
account of Schramm-Loewner evolutions (SLE(k)), a one-parameter family of
conformally invariant random curves discovered by Schramm (2000). The article
is organized around the aim of proving the result, due to Smirnov (2001) and to
Camia and Newman (2007), that the percolation exploration path converges in the
scaling limit to chordal SLE(6). No prior knowledge is assumed beyond some
general complex analysis and probability theory.Comment: 55 pages, 10 figure
M\"obius Invariants of Shapes and Images
Identifying when different images are of the same object despite changes
caused by imaging technologies, or processes such as growth, has many
applications in fields such as computer vision and biological image analysis.
One approach to this problem is to identify the group of possible
transformations of the object and to find invariants to the action of that
group, meaning that the object has the same values of the invariants despite
the action of the group. In this paper we study the invariants of planar shapes
and images under the M\"obius group , which arises
in the conformal camera model of vision and may also correspond to neurological
aspects of vision, such as grouping of lines and circles. We survey properties
of invariants that are important in applications, and the known M\"obius
invariants, and then develop an algorithm by which shapes can be recognised
that is M\"obius- and reparametrization-invariant, numerically stable, and
robust to noise. We demonstrate the efficacy of this new invariant approach on
sets of curves, and then develop a M\"obius-invariant signature of grey-scale
images
Illusions in the Ring Model of visual orientation selectivity
The Ring Model of orientation tuning is a dynamical model of a hypercolumn of
visual area V1 in the human neocortex that has been designed to account for the
experimentally observed orientation tuning curves by local, i.e.,
cortico-cortical computations. The tuning curves are stationary, i.e. time
independent, solutions of this dynamical model. One important assumption
underlying the Ring Model is that the LGN input to V1 is weakly tuned to the
retinal orientation and that it is the local computations in V1 that sharpen
this tuning. Because the equations that describe the Ring Model have built-in
equivariance properties in the synaptic weight distribution with respect to a
particular group acting on the retinal orientation of the stimulus, the model
in effect encodes an infinite number of tuning curves that are arbitrarily
translated with respect to each other. By using the Orbit Space Reduction
technique we rewrite the model equations in canonical form as functions of
polynomials that are invariant with respect to the action of this group. This
allows us to combine equivariant bifurcation theory with an efficient numerical
continuation method in order to compute the tuning curves predicted by the Ring
Model. Surprisingly some of these tuning curves are not tuned to the stimulus.
We interpret them as neural illusions and show numerically how they can be
induced by simple dynamical stimuli. These neural illusions are important
biological predictions of the model. If they could be observed experimentally
this would be a strong point in favour of the Ring Model. We also show how our
theoretical analysis allows to very simply specify the ranges of the model
parameters by comparing the model predictions with published experimental
observations.Comment: 33 pages, 12 figure
Breaking and sustaining bifurcations in SN-Invariant equidistant economy
This paper elucidates the bifurcation mechanism of an equidistant economy in spatial economics. To this end, we derive the rules of secondary and further bifurcations as a major theoretical contribution of this paper. Then we combine them with pre-existing results of direct bifurcation of the symmetric group SN [Elmhirst, 2004]. Particular attention is devoted to the existence of invariant solutions which retain their spatial distributions when the value of the bifurcation parameter changes. Invariant patterns of an equidistant economy under the replicator dynamics are obtained. The mechanism of bifurcations from these patterns is elucidated. The stability of bifurcating branches is analyzed to demonstrate that most of them are unstable immediately after bifurcation. Numerical analysis of spatial economic models confirms that almost all bifurcating branches are unstable. Direct bifurcating curves connect the curves of invariant solutions, thereby creating a mesh-like network, which appears as threads of warp and weft. The theoretical bifurcation mechanism and numerical examples of networks advanced herein might be of great assistance in the study of spatial economics.info:eu-repo/semantics/publishedVersio
On dynamical properties of diffeomorphisms with homoclinic tangencies
We study bifurcations of a homoclinic tangency to a saddle fixed point without non-leading multipliers. We give criteria for the birth of an infinite set of stable periodic orbits, an infinite set of coexisting saddle periodic orbits with different instability indices, non-hyperbolic periodic orbits with more than one multiplier on the unit circle, and an infinite set of stable closed invariant curves (invariant tori). The results are based on the rescaling of the first-return map near the orbit of homoclinic tangency, which is shown to bring the map close to one of four standard quadratic maps, and on the analysis of the bifurcations in these maps
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