20,652 research outputs found
A compact formula for the derivative of a 3-D rotation in exponential coordinates
We present a compact formula for the derivative of a 3-D rotation matrix with
respect to its exponential coordinates. A geometric interpretation of the
resulting expression is provided, as well as its agreement with other
less-compact but better-known formulas. To the best of our knowledge, this
simpler formula does not appear anywhere in the literature. We hope by
providing this more compact expression to alleviate the common pressure to
reluctantly resort to alternative representations in various computational
applications simply as a means to avoid the complexity of differential analysis
in exponential coordinates.Comment: 6 page
A Compact Formula for the Derivative of a 3-D Rotation in Exponential Coordinates
We present a compact formula for the derivative of a 3-D rotation matrix with respect to its exponential coordinates. A geometric interpretation of the resulting expression is provided, as well as its agreement with other less-compact but better-known formulas. To the best of our knowledge, this simpler formula does not appear anywhere in the literature. We hope by providing this more compact expression to alleviate the common pressure to reluctantly resort to alternative representations in various computational applications simply as a means to avoid the complexity of differential analysis in exponential coordinates
Special Lagrangian submanifolds of log Calabi-Yau manifolds
We study the existence of special Lagrangian submanifolds of log Calabi-Yau
manifolds equipped with the complete Ricci-flat K\"ahler metric constructed by
Tian-Yau. We prove that if is a Tian-Yau manifold, and if the compact
Calabi-Yau manifold at infinty admits a single special Lagrangian, then
admits infinitely many disjoint special Lagrangians. In complex dimension ,
we prove that if is a del Pezzo surface, or a rational elliptic surface,
and is a smooth divisor with , then
admits a special Lagrangian torus fibration, as conjectured by
Strominger-Yau-Zaslow and Auroux. In fact, we show that admits twin special
Lagrangian fibrations, confirming a prediction of Leung-Yau. In the special
case that is a rational elliptic surface, or we identify
the singular fibers for generic data, thereby confirming two conjectures of
Auroux. Finally, we prove that after a hyper-K\"ahler rotation, can be
compactified to the complement of a Kodaira type fiber appearing as a
singular fiber in a rational elliptic surface .Comment: 70 pages. Updates and improvements. To appear in Duke Mathematical
Journa
A Primer on the Differential Calculus of 3D Orientations
The proper handling of 3D orientations is a central element in many
optimization problems in engineering. Unfortunately many researchers and
engineers struggle with the formulation of such problems and often fall back to
suboptimal solutions. The existence of many different conventions further
complicates this issue, especially when interfacing multiple differing
implementations. This document discusses an alternative approach which makes
use of a more abstract notion of 3D orientations. The relative orientation
between two coordinate systems is primarily identified by the coordinate
mapping it induces. This is combined with the standard exponential map in order
to introduce representation-independent and minimal differentials, which are
very convenient in optimization based methods
An averaging principle for diffusions in foliated spaces
Consider an SDE on a foliated manifold whose trajectories lay on compact
leaves. We investigate the effective behavior of a small transversal
perturbation of order . An average principle is shown to hold such
that the component transversal to the leaves converges to the solution of a
deterministic ODE, according to the average of the perturbing vector field with
respect to invariant measures on the leaves, as goes to zero. An
estimate of the rate of convergence is given. These results generalize the
geometrical scope of previous approaches, including completely integrable
stochastic Hamiltonian system.Comment: Published at http://dx.doi.org/10.1214/14-AOP982 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The space of essential matrices as a Riemannian quotient manifold
The essential matrix, which encodes the epipolar constraint between points in two projective views,
is a cornerstone of modern computer vision. Previous works have proposed different characterizations
of the space of essential matrices as a Riemannian manifold. However, they either do not consider the
symmetric role played by the two views, or do not fully take into account the geometric peculiarities
of the epipolar constraint. We address these limitations with a characterization as a quotient manifold
which can be easily interpreted in terms of camera poses. While our main focus in on theoretical
aspects, we include applications to optimization problems in computer vision.This work was supported by grants NSF-IIP-0742304, NSF-OIA-1028009, ARL MAST-CTA W911NF-08-2-0004, and ARL RCTA W911NF-10-2-0016, NSF-DGE-0966142, and NSF-IIS-1317788
Special Lagrangian submanifolds of log Calabi-Yau manifolds
We study the existence of special Lagrangian submanifolds of log Calabi-Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian-Yau. We prove that if X is a Tian-Yau manifold, and if the compact Calabi-Yau manifold at infinty admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface, or a rational elliptic surface, and D∈|−KY| is a smooth divisor with D2=d, then X=Y∖D admits a special Lagrangian torus fibration, as conjectured by Strominger-Yau-Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung-Yau. In the special case that Y is a rational elliptic surface, or Y=ℙ2 we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kähler rotation, X can be compactified to the complement of a Kodaira type Id fiber appearing as a singular fiber in a rational elliptic surface πˇ:Yˇ→ℙ1.https://arxiv.org/abs/1904.08363First author draf
Perturbation Theory for the Quantum Time-Evolution in Rotating Potentials
The quantum mechanical time-evolution is studied for a particle under the
influence of an explicitly time-dependent rotating potential. We discuss the
existence of the propagator and we show that in the limit of rapid rotation it
converges strongly to the solution operator of the Schr\"odinger equation with
the averaged rotational invariant potential.Comment: To appear in Proceedings of the Conference QMath-8 "Mathematical
Results in Quantum Mechanics" Taxco, Mexico, December 200
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