557 research outputs found
In Search of Fundamental Discreteness in 2+1 Dimensional Quantum Gravity
Inspired by previous work in 2+1 dimensional quantum gravity, which found
evidence for a discretization of time in the quantum theory, we reexamine the
issue for the case of pure Lorentzian gravity with vanishing cosmological
constant and spatially compact universes of genus larger than 1. Taking as our
starting point the Chern-Simons formulation with Poincare gauge group, we
identify a set of length variables corresponding to space- and timelike
distances along geodesics in three-dimensional Minkowski space. These are Dirac
observables, that is, functions on the reduced phase space, whose quantization
is essentially unique. For both space- and timelike distance operators, the
spectrum is continuous and not bounded away from zero.Comment: 29 pages, 18 figure
Spectral theorems for random walks on mapping class groups and
We establish spectral theorems for random walks on mapping class groups of
connected, closed, oriented, hyperbolic surfaces, and on . In
both cases, we relate the asymptotics of the stretching factor of the
diffeomorphism/automorphism obtained at time of the random walk to the
Lyapunov exponent of the walk, which gives the typical growth rate of the
length of a curve -- or of a conjugacy class in -- under a random product
of diffeomorphisms/automorphisms.
In the mapping class group case, we first observe that the drift of the
random walk in the curve complex is also equal to the linear growth rate of the
translation lengths in this complex. By using a contraction property of typical
Teichm\"uller geodesics, we then lift the above fact to the realization of the
random walk on the Teichm\"uller space. For the case of , we
follow the same procedure with the free factor complex in place of the curve
complex, and the outer space in place of the Teichm\"uller space. A general
criterion is given for making the lifting argument possible.Comment: 45 pages, 3 figures. arXiv admin note: text overlap with
arXiv:1506.0724
Soliton Dynamics in Computational Anatomy
Computational anatomy (CA) has introduced the idea of anatomical structures
being transformed by geodesic deformations on groups of diffeomorphisms. Among
these geometric structures, landmarks and image outlines in CA are shown to be
singular solutions of a partial differential equation that is called the
geodesic EPDiff equation. A recently discovered momentum map for singular
solutions of EPDiff yields their canonical Hamiltonian formulation, which in
turn provides a complete parameterization of the landmarks by their canonical
positions and momenta. The momentum map provides an isomorphism between
landmarks (and outlines) for images and singular soliton solutions of the
EPDiff equation. This isomorphism suggests a new dynamical paradigm for CA, as
well as new data representation.Comment: published in NeuroImag
Multiple Shape Registration using Constrained Optimal Control
Lagrangian particle formulations of the large deformation diffeomorphic
metric mapping algorithm (LDDMM) only allow for the study of a single shape. In
this paper, we introduce and discuss both a theoretical and practical setting
for the simultaneous study of multiple shapes that are either stitched to one
another or slide along a submanifold. The method is described within the
optimal control formalism, and optimality conditions are given, together with
the equations that are needed to implement augmented Lagrangian methods.
Experimental results are provided for stitched and sliding surfaces
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