18,565 research outputs found
The parameterization method for center manifolds
In this paper, we present a generalization of the parameterization method,
introduced by Cabr\'{e}, Fontich and De la Llave, to center manifolds
associated to non-hyperbolic fixed points of discrete dynamical systems. As a
byproduct, we find a new proof for the existence and regularity of center
manifolds. However, in contrast to the classical center manifold theorem, our
parameterization method will simultaneously obtain the center manifold and its
conjugate center dynamical system. Furthermore, we will provide bounds on the
error between approximations of the center manifold and the actual center
manifold, as well as bounds for the error in the conjugate dynamical system
SU(3) Revisited
The ``'' matrices for all states of the two fundamental representations
and octet are shown in the generalized Euler angle parameterization. The
raising and lowering operators are given in terms of linear combinations of the
left invariant vector fields of the group manifold in this parameterization.
Using these differential operators the highest weight state of an arbitrary
irreducible representation is found and a description of the calculation of
Clebsch-Gordon coefficients is given.Comment: 22 pages LaTe
The parameterization method for center manifolds
In this paper, we present a generalization of the parameterization method, introduced by Cabré, Fontich and De la Llave, to center manifolds associated to non-hyperbolic fixed points of discrete dynamical systems. As a byproduct, we find a new proof for the existence and regularity of center manifolds. However, in contrast to the classical center manifold theorem, our parameterization method will simultaneously obtain the center manifold and its conjugate center dynamical system. Furthermore, we will provide bounds on the error between approximations of the center manifold and the actual center manifold, as well as bounds for the error in the conjugate dynamical system
Non-Asymptotic Analysis of Tangent Space Perturbation
Constructing an efficient parameterization of a large, noisy data set of
points lying close to a smooth manifold in high dimension remains a fundamental
problem. One approach consists in recovering a local parameterization using the
local tangent plane. Principal component analysis (PCA) is often the tool of
choice, as it returns an optimal basis in the case of noise-free samples from a
linear subspace. To process noisy data samples from a nonlinear manifold, PCA
must be applied locally, at a scale small enough such that the manifold is
approximately linear, but at a scale large enough such that structure may be
discerned from noise. Using eigenspace perturbation theory and non-asymptotic
random matrix theory, we study the stability of the subspace estimated by PCA
as a function of scale, and bound (with high probability) the angle it forms
with the true tangent space. By adaptively selecting the scale that minimizes
this bound, our analysis reveals an appropriate scale for local tangent plane
recovery. We also introduce a geometric uncertainty principle quantifying the
limits of noise-curvature perturbation for stable recovery. With the purpose of
providing perturbation bounds that can be used in practice, we propose plug-in
estimates that make it possible to directly apply the theoretical results to
real data sets.Comment: 53 pages. Revised manuscript with new content addressing application
of results to real data set
A Girsanov approach to slow parameterizing manifolds in the presence of noise
We consider a three-dimensional slow-fast system with quadratic nonlinearity
and additive noise. The associated deterministic system of this stochastic
differential equation (SDE) exhibits a periodic orbit and a slow manifold. The
deterministic slow manifold can be viewed as an approximate parameterization of
the fast variable of the SDE in terms of the slow variables. In other words the
fast variable of the slow-fast system is approximately "slaved" to the slow
variables via the slow manifold. We exploit this fact to obtain a two
dimensional reduced model for the original stochastic system, which results in
the Hopf-normal form with additive noise. Both, the original as well as the
reduced system admit ergodic invariant measures describing their respective
long-time behaviour. We will show that for a suitable metric on a subset of the
space of all probability measures on phase space, the discrepancy between the
marginals along the radial component of both invariant measures can be upper
bounded by a constant and a quantity describing the quality of the
parameterization. An important technical tool we use to arrive at this result
is Girsanov's theorem, which allows us to modify the SDEs in question in a way
that preserves transition probabilities. This approach is then also applied to
reduced systems obtained through stochastic parameterizing manifolds, which can
be viewed as generalized notions of deterministic slow manifolds.Comment: 54 pages, 6 figure
Path planning for active tensegrity structures
This paper presents a path planning method for actuated tensegrity structures with quasi-static motion. The valid configurations for such structures lay on an equilibrium manifold, which is implicitly defined by a set of kinematic and static constraints. The exploration of this manifold is difficult with standard methods due to the lack of a global parameterization. Thus, this paper proposes the use of techniques with roots in differential geometry to define an atlas, i.e., a set of coordinated local parameterizations of the equilibrium manifold. This atlas is exploited to define a rapidly-exploring random tree, which efficiently finds valid paths between configurations. However, these paths are typically long and jerky and, therefore, this paper also introduces a procedure to reduce their control effort. A variety of test cases are presented to empirically evaluate the proposed method. (C) 2015 Elsevier Ltd. All rights reserved.Peer ReviewedPostprint (author's final draft
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