955 research outputs found
Constructive inversion of energy trajectories in quantum mechanics
We suppose that the ground-state eigenvalue E = F(v) of the Schroedinger
Hamiltonian H = -\Delta + vf(x) in one dimension is known for all values of the
coupling v > 0. The potential shape f(x) is assumed to be symmetric, bounded
below, and monotone increasing for x > 0. A fast algorithm is devised which
allows the potential shape f(x) to be reconstructed from the energy trajectory
F(v). Three examples are discussed in detail: a shifted power-potential, the
exponential potential, and the sech-squared potential are each reconstructed
from their known exact energy trajectories.Comment: 16 pages in plain TeX with 5 ps figure
Inference via low-dimensional couplings
We investigate the low-dimensional structure of deterministic transformations
between random variables, i.e., transport maps between probability measures. In
the context of statistics and machine learning, these transformations can be
used to couple a tractable "reference" measure (e.g., a standard Gaussian) with
a target measure of interest. Direct simulation from the desired measure can
then be achieved by pushing forward reference samples through the map. Yet
characterizing such a map---e.g., representing and evaluating it---grows
challenging in high dimensions. The central contribution of this paper is to
establish a link between the Markov properties of the target measure and the
existence of low-dimensional couplings, induced by transport maps that are
sparse and/or decomposable. Our analysis not only facilitates the construction
of transformations in high-dimensional settings, but also suggests new
inference methodologies for continuous non-Gaussian graphical models. For
instance, in the context of nonlinear state-space models, we describe new
variational algorithms for filtering, smoothing, and sequential parameter
inference. These algorithms can be understood as the natural
generalization---to the non-Gaussian case---of the square-root
Rauch-Tung-Striebel Gaussian smoother.Comment: 78 pages, 25 figure
Functional co-monotony of processes with applications to peacocks and barrier options
We show that several general classes of stochastic processes satisfy a
functional co-monotony principle, including processes with independent
increments, Brownian diffusions, Liouville processes. As a first application,
we recover some recent results about peacock processes obtained by Hirsch et
al. which were themselves motivated by a former work of Carr et al. about the
sensitivity of Asian Call options with respect to their volatility and residual
maturity (seniority). We also derive semi-universal bounds for various barrier
options.Comment: 27 page
Prabhakar-like fractional viscoelasticity
The aim of this paper is to present a linear viscoelastic model based on
Prabhakar fractional operators. In particular, we propose a modification of the
classical fractional Maxwell model, in which we replace the Caputo derivative
with the Prabhakar one. Furthermore, we also discuss how to recover a formal
equivalence between the new model and the known classical models of linear
viscoelasticity by means of a suitable choice of the parameters in the
Prabhakar derivative. Moreover, we also underline an interesting connection
between the theory of Prabhakar fractional integrals and the recently
introduced Caputo-Fabrizio differential operator.Comment: 9 page
Estimation of a -monotone density: limit distribution theory and the spline connection
We study the asymptotic behavior of the Maximum Likelihood and Least Squares
Estimators of a -monotone density at a fixed point when .
We find that the th derivative of the estimators at converges at the
rate for . The limiting distribution depends
on an almost surely uniquely defined stochastic process that stays above
(below) the -fold integral of Brownian motion plus a deterministic drift
when is even (odd). Both the MLE and LSE are known to be splines of degree
with simple knots. Establishing the order of the random gap
, where denote two successive knots, is a key
ingredient of the proof of the main results. We show that this ``gap problem''
can be solved if a conjecture about the upper bound on the error in a
particular Hermite interpolation via odd-degree splines holds.Comment: Published in at http://dx.doi.org/10.1214/009053607000000262 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Generalizations of Kochen and Specker's Theorem and the Effectiveness of Gleason's Theorem
Kochen and Specker's theorem can be seen as a consequence of Gleason's
theorem and logical compactness. Similar compactness arguments lead to stronger
results about finite sets of rays in Hilbert space, which we also prove by a
direct construction. Finally, we demonstrate that Gleason's theorem itself has
a constructive proof, based on a generic, finite, effectively generated set of
rays, on which every quantum state can be approximated.Comment: 14 pages, 6 figures, read at the Robert Clifton memorial conferenc
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