102,390 research outputs found
Map Matching with Simplicity Constraints
We study a map matching problem, the task of finding in an embedded graph a
path that has low distance to a given curve in R^2. The Fr\'echet distance is a
common measure for this problem. Efficient methods exist to compute the best
path according to this measure. However, these methods cannot guarantee that
the result is simple (i.e. it does not intersect itself) even if the given
curve is simple. In this paper, we prove that it is in fact NP-complete to
determine the existence a simple cycle in a planar straight-line embedding of a
graph that has at most a given Fr\'echet distance to a given simple closed
curve. We also consider the implications of our proof on some variants of the
problem
New Variables for Classical and Quantum Gravity in all Dimensions II. Lagrangian Analysis
We rederive the results of our companion paper, for matching spacetime and
internal signature, by applying in detail the Dirac algorithm to the Palatini
action. While the constraint set of the Palatini action contains second class
constraints, by an appeal to the method of gauge unfixing, we map the second
class system to an equivalent first class system which turns out to be
identical to the first class constraint system obtained via the extension of
the ADM phase space performed in our companion paper. Central to our analysis
is again the appropriate treatment of the simplicity constraint. Remarkably,
the simplicity constraint invariant extension of the Hamiltonian constraint,
that is a necessary step in the gauge unfixing procedure, involves a correction
term which is precisely the one found in the companion paper and which makes
sure that the Hamiltonian constraint derived from the Palatini Lagrangian
coincides with the ADM Hamiltonian constraint when Gauss and simplicity
constraints are satisfied. We therefore have rederived our new connection
formulation of General Relativity from an independent starting point, thus
confirming the consistency of this framework.Comment: 42 pages. v2: Journal version. Some nonessential sign errors in
section 2 corrected. Minor clarification
Lifting SU(2) Spin Networks to Projected Spin Networks
Projected spin network states are the canonical basis of quantum states of
geometry for the most recent EPR-FK spinfoam models for quantum gravity. They
are functionals of both the Lorentz connection and the time normal field. We
analyze in details the map from these projected spin networks to the standard
SU(2) spin networks of loop quantum gravity. We show that this map is not
one-to-one and that the corresponding ambiguity is parameterized by the Immirzi
parameter. We conclude with a comparison of the scalar products between
projected spin networks and SU(2) spin network states.Comment: 14 page
U(N) and holomorphic methods for LQG and Spin Foams
The U(N) framework and the spinor representation for loop quantum gravity are
two new points of view that can help us deal with the most fundamental problems
of the theory. Here, we review the detailed construction of the U(N) framework
explaining how one can endow the Hilbert space of N-leg intertwiners with a
Fock structure. We then give a description of the classical phase space
corresponding to this system in terms of the spinors, and we will study its
quantization using holomorphic techniques. We take special care in constructing
the usual holonomy operators of LQG in terms of spinors, and in the description
of the Hilbert space of LQG with the different polarization given by these
spinors.Comment: 16 pages. Proceedings for the 3rd Quantum Geometry and Quantum
Gravity School in Zakopane (2011
Classical Setting and Effective Dynamics for Spinfoam Cosmology
We explore how to extract effective dynamics from loop quantum gravity and
spinfoams truncated to a finite fixed graph, with the hope of modeling
symmetry-reduced gravitational systems. We particularize our study to the
2-vertex graph with N links. We describe the canonical data using the recent
formulation of the phase space in terms of spinors, and implement a
symmetry-reduction to the homogeneous and isotropic sector. From the canonical
point of view, we construct a consistent Hamiltonian for the model and discuss
its relation with Friedmann-Robertson-Walker cosmologies. Then, we analyze the
dynamics from the spinfoam approach. We compute exactly the transition
amplitude between initial and final coherent spin networks states with support
on the 2-vertex graph, for the choice of the simplest two-complex (with a
single space-time vertex). The transition amplitude verifies an exact
differential equation that agrees with the Hamiltonian constructed previously.
Thus, in our simple setting we clarify the link between the canonical and the
covariant formalisms.Comment: 38 pages, v2: Link with discretized loop quantum gravity made
explicit and emphasize
Null twisted geometries
We define and investigate a quantisation of null hypersurfaces in the context
of loop quantum gravity on a fixed graph. The main tool we use is the
parametrisation of the theory in terms of twistors, which has already proved
useful in discussing the interpretation of spin networks as the quantization of
twisted geometries. The classical formalism can be extended in a natural way to
null hypersurfaces, with the Euclidean polyhedra replaced by null polyhedra
with space-like faces, and SU(2) by the little group ISO(2). The main
difference is that the simplicity constraints present in the formalims are all
first class, and the symplectic reduction selects only the helicity subgroup of
the little group. As a consequence, information on the shapes of the polyhedra
is lost, and the result is a much simpler, abelian geometric picture. It can be
described by an Euclidean singular structure on the 2-dimensional space-like
surface defined by a foliation of space-time by null hypersurfaces. This
geometric structure is naturally decomposed into a conformal metric and scale
factors, forming locally conjugate pairs. Proper action-angle variables on the
gauge-invariant phase space are described by the eigenvectors of the Laplacian
of the dual graph. We also identify the variables of the phase space amenable
to characterize the extrinsic geometry of the foliation. Finally, we quantise
the phase space and its algebra using Dirac's algorithm, obtaining a notion of
spin networks for null hypersurfaces. Such spin networks are labelled by SO(2)
quantum numbers, and are embedded non-trivially in the unitary,
infinite-dimensional irreducible representations of the Lorentz group.Comment: 22 pages, 3 figures. v2: minor corrections, improved presentation in
section 4, references update
The edge cloud: A holistic view of communication, computation and caching
The evolution of communication networks shows a clear shift of focus from
just improving the communications aspects to enabling new important services,
from Industry 4.0 to automated driving, virtual/augmented reality, Internet of
Things (IoT), and so on. This trend is evident in the roadmap planned for the
deployment of the fifth generation (5G) communication networks. This ambitious
goal requires a paradigm shift towards a vision that looks at communication,
computation and caching (3C) resources as three components of a single holistic
system. The further step is to bring these 3C resources closer to the mobile
user, at the edge of the network, to enable very low latency and high
reliability services. The scope of this chapter is to show that signal
processing techniques can play a key role in this new vision. In particular, we
motivate the joint optimization of 3C resources. Then we show how graph-based
representations can play a key role in building effective learning methods and
devising innovative resource allocation techniques.Comment: to appear in the book "Cooperative and Graph Signal Pocessing:
Principles and Applications", P. Djuric and C. Richard Eds., Academic Press,
Elsevier, 201
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