820 research outputs found
Quasi-convergence of an implementation of optimal balance by backward-forward nudging
Optimal balance is a non-asymptotic numerical method to compute a point on
the slow manifold for certain two-scale dynamical systems. It works by solving
a modified version of the system as a boundary value problem in time, where the
nonlinear terms are adiabatically ramped up from zero to the fully nonlinear
dynamics. A dedicated boundary value solver, however, is often not directly
available. The most natural alternative is a nudging solver, where the problem
is repeatedly solved forward and backward in time and the respective boundary
conditions are restored whenever one of the temporal end points is visited. In
this paper, we show quasi-convergence of this scheme in the sense that the
termination residual of the nudging iteration is as small as the asymptotic
error of the method itself, i.e., under appropriate assumptions exponentially
small. This confirms that optimal balance in its nudging formulation is an
effective algorithm. Further, it shows that the boundary value problem
formulation of optimal balance is well posed up at most a residual error as
small as the asymptotic error of the method itself. The key step in our proof
is a careful two-component Gronwall inequality
Length functions on currents and applications to dynamics and counting
The aim of this (mostly expository) article is twofold. We first explore a
variety of length functions on the space of currents, and we survey recent work
regarding applications of length functions to counting problems. Secondly, we
use length functions to provide a proof of a folklore theorem which states that
pseudo-Anosov homeomorphisms of closed hyperbolic surfaces act on the space of
projective geodesic currents with uniform north-south dynamics.Comment: 35pp, 2 figures, comments welcome! Second version: minor corrections.
To appear as a chapter in the forthcoming book "In the tradition of Thurston"
edited by V. Alberge, K. Ohshika and A. Papadopoulo
Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow
We compute the sum of the positive Lyapunov exponents of the Hodge bundle
with respect to the Teichmuller geodesic flow. The computation is based on the
analytic Riemann-Roch Theorem and uses a comparison of determinants of flat and
hyperbolic Laplacians when the underlying Riemann surface degenerates.Comment: Minor corrections. To appear in Publications mathematiques de l'IHE
Comment on "c-axis Josephson tunneling in -wave superconductors''
This comment points out that the recent paper by Maki and Haas [Phys. Rev. B
{\bf 67}, 020510 (2003)] is completely wrong.Comment: 1 page, submittted to Phys. Rev.
Square-tiled cyclic covers
A cyclic cover of the complex projective line branched at four appropriate
points has a natural structure of a square-tiled surface. We describe the
combinatorics of such a square-tiled surface, the geometry of the corresponding
Teichm\"uller curve, and compute the Lyapunov exponents of the determinant
bundle over the Teichm\"uller curve with respect to the geodesic flow. This
paper includes a new example (announced by G. Forni and C. Matheus in
\cite{Forni:Matheus}) of a Teichm\"uller curve of a square-tiled cyclic cover
in a stratum of Abelian differentials in genus four with a maximally degenerate
Kontsevich--Zorich spectrum (the only known example found previously by Forni
in genus three also corresponds to a square-tiled cyclic cover
\cite{ForniSurvey}).
We present several new examples of Teichm\"uller curves in strata of
holomorphic and meromorphic quadratic differentials with maximally degenerate
Kontsevich--Zorich spectrum. Presumably, these examples cover all possible
Teichm\"uller curves with maximally degenerate spectrum. We prove that this is
indeed the case within the class of square-tiled cyclic covers.Comment: 34 pages, 6 figures. Final version incorporating referees comments.
In particular, a gap in the previous version was corrected. This file uses
the journal's class file (jmd.cls), so that it is very similar to published
versio
Geometric Aspects of the Moduli Space of Riemann Surfaces
This is a survey of our recent results on the geometry of moduli spaces and
Teichmuller spaces of Riemann surfaces appeared in math.DG/0403068 and
math.DG/0409220. We introduce new metrics on the moduli and the Teichmuller
spaces of Riemann surfaces with very good properties, study their curvatures
and boundary behaviors in great detail. Based on the careful analysis of these
new metrics, we have a good understanding of the Kahler-Einstein metric from
which we prove that the logarithmic cotangent bundle of the moduli space is
stable. Another corolary is a proof of the equivalences of all of the known
classical complete metrics to the new metrics, in particular Yau's conjectures
in the early 80s on the equivalences of the Kahler-Einstein metric to the
Teichmuller and the Bergman metric.Comment: Survey article of our recent results on the subject. Typoes
corrrecte
Quadratic differentials as stability conditions
We prove that moduli spaces of meromorphic quadratic differentials with
simple zeroes on compact Riemann surfaces can be identified with spaces of
stability conditions on a class of CY3 triangulated categories defined using
quivers with potential associated to triangulated surfaces. We relate the
finite-length trajectories of such quadratic differentials to the stable
objects of the corresponding stability condition.Comment: 123 pages; 38 figures. Version 2: hypotheses in the main results
mildly weakened, to reflect improved results of Labardini-Fragoso and
coauthors. Version 3: minor changes to incorporate referees' suggestions.
This version to appear in Publ. Math. de l'IHE
Magnetisation configuration in arrays of permalloy rectangles and its impact on magnetisation reversal
The remanent domain structures of composite element magnetic barcodes have been imaged using photo-emission electron microscopy with contrast from x-ray magnetic circular dichroism (XMCD-PEEM) and analysed with reference to the results of micromagnetic simulations. The magnetisation configuration at the end of wide strips is found to be perpendicular to the majority magnetisation direction. This transitions to an incomplete rotation for nominal strip widths below 300 nm and is found to affect the mechanics of magnetisation reversal for nominal strip widths below 200 nm, owing to a difference in magnetisation orientation when an external magnetic field is applied that is just smaller than the magnetic coercivity of the structures and a corresponding change in reversal dynamics. This change in domain structure as strip width decreases is consistent with both the influence of shape anisotropy and with measurements of magnetic hysteresis. The magnetisation reversal characteristics of composite element structures are found to be dependent on the relative magnetisation configurations of neighbouring strips, which in turn are found to vary stochastically upon the application and removal of a magnetic field along the easy axis of the structure. It is found that the application of a canted field is necessary to ensure sharp, consistent magnetisation reversal of bits when writing a binary code. These results confirm that either improved lithography of narrower strips or non-rectangular elements would be necessary to further increase the number of individually programmable bits in a barcode
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