103 research outputs found
Averaged Template Matching Equations
By exploiting an analogy with averaging procedures in fluid
dynamics, we present a set of averaged template matching equations.
These equations are analogs of the exact template matching equations
that retain all the geometric properties associated with the diffeomorphismgrou
p, and which are expected to average out small scale features
and so should, as in hydrodynamics, be more computationally efficient
for resolving the larger scale features. Froma geometric point of view,
the new equations may be viewed as coming from a change in norm that
is used to measure the distance between images. The results in this paper
represent first steps in a longer termpro gram: what is here is only
for binary images and an algorithm for numerical computation is not
yet operational. Some suggestions for further steps to develop the results
given in this paper are suggested
The Newtonian Limit for Asymptotically Flat Solutions of the Vlasov-Einstein System
It is shown that there exist families of asymptotically flat solutions of the
Einstein equations coupled to the Vlasov equation describing a collisionless
gas which have a Newtonian limit. These are sufficiently general to confirm
that for this matter model as many families of this type exist as would be
expected on the basis of physical intuition. A central role in the proof is
played by energy estimates in unweighted Sobolev spaces for a wave equation
satisfied by the second fundamental form of a maximal foliation.Comment: 24 pages, plain TE
An Introduction to Conformal Ricci Flow
We introduce a variation of the classical Ricci flow equation that modifies
the unit volume constraint of that equation to a scalar curvature constraint.
The resulting equations are named the Conformal Ricci Flow Equations because of
the role that conformal geometry plays in constraining the scalar curvature.
These equations are analogous to the incompressible Navier-Stokes equations of
fluid mechanics inasmuch as a conformal pressure arises as a Lagrange
multiplier to conformally deform the metric flow so as to maintain the scalar
curvature constraint. The equilibrium points are Einstein metrics with a
negative Einstein constant and the conformal pressue is shown to be zero at an
equilibrium point and strictly positive otherwise. The geometry of the
conformal Ricci flow is discussed as well as the remarkable analytic fact that
the constraint force does not lose derivatives and thus analytically the
conformal Ricci equation is a bounded perturbation of the classical
unnormalized Ricci equation. That the constraint force does not lose
derivatives is exactly analogous to the fact that the real physical pressure
force that occurs in the Navier-Stokes equations is a bounded function of the
velocity. Using a nonlinear Trotter product formula, existence and uniqueness
of solutions to the conformal Ricci flow equations is proven. Lastly, we
discuss potential applications to Perelman's proposed implementation of
Hamilton's program to prove Thurston's 3-manifold geometrization conjectures.Comment: 52 pages, 1 figur
Toeplitz Quantization of K\"ahler Manifolds and
For general compact K\"ahler manifolds it is shown that both Toeplitz
quantization and geometric quantization lead to a well-defined (by operator
norm estimates) classical limit. This generalizes earlier results of the
authors and Klimek and Lesniewski obtained for the torus and higher genus
Riemann surfaces, respectively. We thereby arrive at an approximation of the
Poisson algebra by a sequence of finite-dimensional matrix algebras ,
.Comment: 17 pages, AmsTeX 2.1, Sept. 93 (rev: only typos are corrected
The Dynamics of a Rigid Body in Potential Flow with Circulation
We consider the motion of a two-dimensional body of arbitrary shape in a
planar irrotational, incompressible fluid with a given amount of circulation
around the body. We derive the equations of motion for this system by
performing symplectic reduction with respect to the group of volume-preserving
diffeomorphisms and obtain the relevant Poisson structures after a further
Poisson reduction with respect to the group of translations and rotations. In
this way, we recover the equations of motion given for this system by Chaplygin
and Lamb, and we give a geometric interpretation for the Kutta-Zhukowski force
as a curvature-related effect. In addition, we show that the motion of a rigid
body with circulation can be understood as a geodesic flow on a central
extension of the special Euclidian group SE(2), and we relate the cocycle in
the description of this central extension to a certain curvature tensor.Comment: 28 pages, 2 figures; v2: typos correcte
Discrete exterior calculus (DEC) for the surface Navier-Stokes equation
We consider a numerical approach for the incompressible surface Navier-Stokes
equation. The approach is based on the covariant form and uses discrete
exterior calculus (DEC) in space and a semi-implicit discretization in time.
The discretization is described in detail and related to finite difference
schemes on staggered grids in flat space for which we demonstrate second order
convergence. We compare computational results with a vorticity-stream function
approach for surfaces with genus 0 and demonstrate the interplay between
topology, geometry and flow properties. Our discretization also allows to
handle harmonic vector fields, which we demonstrate on a torus.Comment: 21 pages, 9 figure
Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle
In this paper, we study the geodesic flow of a right-invariant metric induced
by a general Fourier multiplier on the diffeomorphism group of the circle and
on some of its homogeneous spaces. This study covers in particular
right-invariant metrics induced by Sobolev norms of fractional order. We show
that, under a certain condition on the symbol of the inertia operator (which is
satisfied for the fractional Sobolev norm for ), the
corresponding initial value problem is well-posed in the smooth category and
that the Riemannian exponential map is a smooth local diffeomorphism.
Paradigmatic examples of our general setting cover, besides all traditional
Euler equations induced by a local inertia operator, the Constantin-Lax-Majda
equation, and the Euler-Weil-Petersson equation.Comment: 40 pages. Corrected typos and improved redactio
The geometry of a vorticity model equation
We provide rigorous evidence of the fact that the modified
Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics
describes the geodesic flow on the subgroup of orientation-preserving
diffeomorphisms fixing one point, with respect to right-invariant metric
induced by the homogeneous Sobolev norm and show the local existence
of the geodesics in the extended group of diffeomorphisms of Sobolev class
with .Comment: 24 page
Interaction of vortices in viscous planar flows
We consider the inviscid limit for the two-dimensional incompressible
Navier-Stokes equation in the particular case where the initial flow is a
finite collection of point vortices. We suppose that the initial positions and
the circulations of the vortices do not depend on the viscosity parameter \nu,
and we choose a time T > 0 such that the Helmholtz-Kirchhoff point vortex
system is well-posed on the interval [0,T]. Under these assumptions, we prove
that the solution of the Navier-Stokes equation converges, as \nu -> 0, to a
superposition of Lamb-Oseen vortices whose centers evolve according to a
viscous regularization of the point vortex system. Convergence holds uniformly
in time, in a strong topology which allows to give an accurate description of
the asymptotic profile of each individual vortex. In particular, we compute to
leading order the deformations of the vortices due to mutual interactions. This
allows to estimate the self-interactions, which play an important role in the
convergence proof.Comment: 39 pages, 1 figur
Normal families of functions and groups of pseudoconformal diffeomorphisms of quaternion and octonion variables
This paper is devoted to the specific class of pseudoconformal mappings of
quaternion and octonion variables. Normal families of functions are defined and
investigated. Four criteria of a family being normal are proven. Then groups of
pseudoconformal diffeomorphisms of quaternion and octonion manifolds are
investigated. It is proven, that they are finite dimensional Lie groups for
compact manifolds. Their examples are given. Many charactersitic features are
found in comparison with commutative geometry over or .Comment: 55 pages, 53 reference
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