2,433 research outputs found
Topological complexity of motion planning in projective product spaces
We study Farber's topological complexity (TC) of Davis' projective product
spaces (PPS's). We show that, in many non-trivial instances, the TC of PPS's
coming from at least two sphere factors is (much) lower than the dimension of
the manifold. This is in high contrast with the known situation for (usual)
real projective spaces for which, in fact, the Euclidean immersion dimension
and TC are two facets of the same problem. Low TC-values have been observed for
infinite families of non-simply connected spaces only for H-spaces, for finite
complexes whose fundamental group has cohomological dimension not exceeding 2,
and now in this work for infinite families of PPS's. We discuss general bounds
for the TC (and the Lusternik-Schnirelmann category) of PPS's, and compute
these invariants for specific families of such manifolds. Some of our methods
involve the use of an equivariant version of TC. We also give a
characterization of the Euclidean immersion dimension of PPS's through
generalized concepts of axial maps and, alternatively, non-singular maps. This
gives an explicit explanation of the known relationship between the generalized
vector field problem and the Euclidean immersion problem for PPS's.Comment: 16 page
Radon Numbers for Trees
Many interesting problems are obtained by attempting to generalize classical
results on convexity in Euclidean spaces to other convexity spaces, in
particular to convexity spaces on graphs. In this paper we consider
-convexity on graphs. A set of vertices in a graph is -convex
if every vertex not in has at most one neighbour in . More specifically,
we consider Radon numbers for -convexity in trees.
Tverberg's theorem states that every set of points in
can be partitioned into sets with intersecting convex hulls.
As a special case of Eckhoff's conjecture, we show that a similar result holds
for -convexity in trees.
A set of vertices in a graph is called free, if no vertex of has
more than one neighbour in . We prove an inequality relating the Radon
number for -convexity in trees with the size of a maximal free set.Comment: 17 pages, 13 figure
Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data
We estimate the quantum state of a light beam from results of quantum
homodyne measurements performed on identically prepared quantum systems. The
state is represented through the Wigner function, a generalized probability
density on which may take negative values and must respect
intrinsic positivity constraints imposed by quantum physics. The effect of the
losses due to detection inefficiencies, which are always present in a real
experiment, is the addition to the tomographic data of independent Gaussian
noise. We construct a kernel estimator for the Wigner function, prove that it
is minimax efficient for the pointwise risk over a class of infinitely
differentiable functions, and implement it for numerical results. We construct
adaptive estimators, that is, which do not depend on the smoothness parameters,
and prove that in some setups they attain the minimax rates for the
corresponding smoothness class.Comment: Published at http://dx.doi.org/10.1214/009053606000001488 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Variational approximation of functionals defined on 1-dimensional connected sets: the planar case
In this paper we consider variational problems involving 1-dimensional
connected sets in the Euclidean plane, such as the classical Steiner tree
problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal
partition problems and provide a variational approximation through
Modica-Mortola type energies proving a -convergence result. We also
introduce a suitable convex relaxation and develop the corresponding numerical
implementations. The proposed methods are quite general and the results we
obtain can be extended to -dimensional Euclidean space or to more general
manifold ambients, as shown in the companion paper [11].Comment: 30 pages, 5 figure
Minimax estimation of the Wigner function in quantum homodyne tomography with ideal detectors
We estimate the quantum state of a light beam from results of quantum
homodyne measurements performed on identically prepared pulses. The state is
represented through the Wigner function, a ``quasi-probability density'' on
which may take negative values and must respect intrinsic
positivity constraints imposed by quantum physics. The data consists of
i.i.d. observations from a probability density equal to the Radon transform of
the Wigner function. We construct an estimator for the Wigner function, and
prove that it is minimax efficient for the pointwise risk over a class of
infinitely differentiable functions. A similar result was previously derived by
Cavalier in the context of positron emission tomography. Our work extends this
result to the space of smooth Wigner functions, which is the relevant parameter
space for quantum homodyne tomography.Comment: 15 page
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