31,436 research outputs found
The problem of retraction in critical discussion
The problem is to find a model of dialogue that allows retractions where they seem reasonable or even required, and puts sanctions on them (or even bans them altogether) whenever they would be disruptive of a well-organized process of dialogue. One ty pe of solution will let retraction rules determine which retractions are permissible, and if permissible what the consequences of retraction are. These rules vary according to the type of dialogue and to the type of commitment to which the retraction per tains. To accommodate various incoherent intuitions on retractions, one may resort to modelling complex types of dialogue
Global rates of convergence for nonconvex optimization on manifolds
We consider the minimization of a cost function on a manifold using
Riemannian gradient descent and Riemannian trust regions (RTR). We focus on
satisfying necessary optimality conditions within a tolerance .
Specifically, we show that, under Lipschitz-type assumptions on the pullbacks
of to the tangent spaces of , both of these algorithms produce points
with Riemannian gradient smaller than in
iterations. Furthermore, RTR returns a point where also the Riemannian
Hessian's least eigenvalue is larger than in
iterations. There are no assumptions on initialization.
The rates match their (sharp) unconstrained counterparts as a function of the
accuracy (up to constants) and hence are sharp in that sense.
These are the first deterministic results for global rates of convergence to
approximate first- and second-order Karush-Kuhn-Tucker points on manifolds.
They apply in particular for optimization constrained to compact submanifolds
of , under simpler assumptions.Comment: 33 pages, IMA Journal of Numerical Analysis, 201
The motion, stability and breakup of a stretching liquid bridge with a receding contact line
The complex behavior of drop deposition on a hydrophobic surface is
considered by looking at a model problem in which the evolution of a
constant-volume liquid bridge is studied as the bridge is stretched. The bridge
is pinned with a fixed diameter at the upper contact point, but the contact
line at the lower attachment point is free to move on a smooth substrate.
Experiments indicate that initially, as the bridge is stretched, the lower
contact line slowly retreats inwards. However at a critical radius, the bridge
becomes unstable, and the contact line accelerates dramatically, moving inwards
very quickly. The bridge subsequently pinches off, and a small droplet is left
on the substrate. A quasi-static analysis, using the Young-Laplace equation, is
used to accurately predict the shape of the bridge during the initial bridge
evolution, including the initial onset of the slow contact line retraction. A
stability analysis is used to predict the onset of pinch-off, and a
one-dimensional dynamical equation, coupled with a Tanner-law for the dynamic
contact angle, is used to model the rapid pinch-off behavior. Excellent
agreement between numerical predictions and experiments is found throughout the
bridge evolution, and the importance of the dynamic contact line model is
demonstrated.Comment: 37 pages, 12 figure
Polyfolds: A First and Second Look
Polyfold theory was developed by Hofer-Wysocki-Zehnder by finding
commonalities in the analytic framework for a variety of geometric elliptic
PDEs, in particular moduli spaces of pseudoholomorphic curves. It aims to
systematically address the common difficulties of compactification and
transversality with a new notion of smoothness on Banach spaces, new local
models for differential geometry, and a nonlinear Fredholm theory in the new
context. We shine meta-mathematical light on the bigger picture and core ideas
of this theory. In addition, we compiled and condensed the core definitions and
theorems of polyfold theory into a streamlined exposition, and outline their
application at the example of Morse theory.Comment: 62 pages, 2 figures. Example 2.1.3 has been modified. Final version,
to appear in the EMS Surv. Math. Sc
Well-rounded equivariant deformation retracts of Teichm\"uller spaces
In this paper, we construct spines, i.e., \Mod_g-equivariant deformation
retracts, of the Teichm\"uller space \T_g of compact Riemann surfaces of
genus . Specifically, we define a \Mod_g-stable subspace of positive
codimension and construct an intrinsic \Mod_g-equivariant deformation
retraction from \T_g to . As an essential part of the proof, we construct
a canonical \Mod_g-deformation retraction of the Teichm\"uller space \T_g
to its thick part \T_g(\varepsilon) when is sufficiently small.
These equivariant deformation retracts of \T_g give cocompact models of the
universal space \underline{E}\Mod_g for proper actions of the mapping class
group \Mod_g. These deformation retractions of \T_g are motivated by the
well-rounded deformation retraction of the space of lattices in . We also
include a summary of results and difficulties of an unpublished paper of
Thurston on a potential spine of the Teichm\"uller space.Comment: A revised version. L'Enseignement Mathematique, 201
Convergence analysis of Riemannian Gauss-Newton methods and its connection with the geometric condition number
We obtain estimates of the multiplicative constants appearing in local
convergence results of the Riemannian Gauss-Newton method for least squares
problems on manifolds and relate them to the geometric condition number of [P.
B\"urgisser and F. Cucker, Condition: The Geometry of Numerical Algorithms,
2013]
- …