2,188 research outputs found
Q-learning with censored data
We develop methodology for a multistage decision problem with flexible number
of stages in which the rewards are survival times that are subject to
censoring. We present a novel Q-learning algorithm that is adjusted for
censored data and allows a flexible number of stages. We provide finite sample
bounds on the generalization error of the policy learned by the algorithm, and
show that when the optimal Q-function belongs to the approximation space, the
expected survival time for policies obtained by the algorithm converges to that
of the optimal policy. We simulate a multistage clinical trial with flexible
number of stages and apply the proposed censored-Q-learning algorithm to find
individualized treatment regimens. The methodology presented in this paper has
implications in the design of personalized medicine trials in cancer and in
other life-threatening diseases.Comment: Published in at http://dx.doi.org/10.1214/12-AOS968 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Dispersive Estimates for higher dimensional Schr\"odinger Operators with threshold eigenvalues II: The even dimensional case
We investigate dispersive
estimates for the Schr\"odinger operator when there is an
eigenvalue at zero energy in even dimensions . In particular, we show
that if there is an eigenvalue at zero energy then there is a time dependent,
rank one operator satisfying for such that With
stronger decay conditions on the potential it is possible to generate an
operator-valued expansion for the evolution, taking the form \begin{align*}
e^{itH} P_{ac}(H)=|t|^{2-\frac{n}{2}}A_{-2}+ |t|^{1-\frac{n}{2}}
A_{-1}+|t|^{-\frac{n}{2}}A_0, \end{align*} with and mapping
to while maps weighted
spaces to weighted spaces. The leading-order terms and
are both finite rank, and vanish when certain orthogonality conditions
between the potential and the zero energy eigenfunctions are satisfied. We
show that under the same orthogonality conditions, the remaining
term also exists as a map from to
, hence satisfies the same dispersive
bounds as the free evolution despite the eigenvalue at zero.Comment: This article continues the work of "Dispersive Estimates for higher
dimensional Schr\"odinger Operators with threshold eigenvalues I: The odd
dimensional case" by the authors to the case of even dimensions. To appear in
J. Spectr. Theor
The boundedness of wave operators for Schr\"odinger Operators with threshold singularities
Let be a Schr\"odinger operator on with
real-valued potential for and let . If decays
sufficiently, the wave operators are known to be bounded on for all if zero is not an eigenvalue, and on if zero is
an eigenvalue. We show that these wave operators are also bounded on
by direct examination of the integral kernel of the leading
term. Furthermore, if for all
eigenfunctions , then the wave operators are bounded for . If, in addition , then the wave
operators are bounded for .Comment: Incorporated referee comments and updated references. To appear in
Adv. Mat
Dispersive Estimates for higher dimensional Schr\"odinger Operators with threshold eigenvalues I: The odd dimensional case
We investigate dispersive
estimates for the Schr\"odinger operator when there is an
eigenvalue at zero energy and is odd. In particular, we show that if
there is an eigenvalue at zero energy then there is a time dependent, rank one
operator satisfying for such that With
stronger decay conditions on the potential it is possible to generate an
operator-valued expansion for the evolution, taking the form with and
finite rank operators mapping to while maps weighted spaces to weighted spaces. The
leading order terms and vanish when certain orthogonality
conditions between the potential and the zero energy eigenfunctions are
satisfied. We show that under the same orthogonality conditions, the remaining
term also exists as a map from to
, hence satisfies the same dispersive
bounds as the free evolution despite the eigenvalue at zero.Comment: To appear in J. Funct. Ana
Limiting absorption principle and Strichartz estimates for Dirac operators in two and higher dimensions
In this paper we consider Dirac operators in , , with a
potential . Under mild decay and continuity assumptions on and some
spectral assumptions on the operator, we prove a limiting absorption principle
for the resolvent, which implies a family of Strichartz estimates for the
linear Dirac equation. For large potentials the dynamical estimates are not an
immediate corollary of the free case since the resolvent of the free Dirac
operator does not decay in operator norm on weighted spaces as the
frequency goes to infinity.Comment: Updated Corollary 1.3 with a slightly stronger statement. To appear
in Comm. Math. Phys. arXiv admin note: text overlap with arXiv:0705.054
Dispersive estimates for four dimensional Schr\"{o}dinger and wave equations with obstructions at zero energy
We investigate dispersive
estimates for the Schr\"odinger operator when there are
obstructions, a resonance or an eigenvalue, at zero energy. In particular, we
show that if there is a resonance or an eigenvalue at zero energy then there is
a time dependent, finite rank operator satisfying for such that
We also show that the operator if there is an eigenvalue but no
resonance at zero energy. We then develop analogous dispersive estimates for
the solution operator to the four dimensional wave equation with potential.Comment: 32 page
Time integrable weighted dispersive estimates for the fourth order Schr\"odinger equation in three dimensions
We consider the fourth order Schr\"odinger operator and show
that if there are no eigenvalues or resonances in the absolutely continuous
spectrum of that the solution operator satisfies a large time
integrable decay rate between weighted spaces. This bound
improves what is possible for the free case in two directions; both better time
decay and smaller spatial weights. In the case of a mild resonance at zero
energy, we derive the operator-valued expansion where is an operator of rank at most
four and maps between polynomially weighted spaces.Comment: 24 pages, submitted. Revised according to referee's comments. arXiv
admin note: text overlap with arXiv:1905.0289
On the boundedness of wave operators for two-dimensional Schr\"odinger operators with threshold obstructions
Let be a Schr\"odinger operator on with
real-valued potential , and let . If has sufficient
pointwise decay, the wave operators are known to be bounded on for all if zero is not an eigenvalue or resonance. We show that if there is an
s-wave resonance or an eigenvalue only at zero, then the wave operators are
bounded on for . This result stands in
contrast to results in higher dimensions, where the presence of zero energy
obstructions is known to shrink the range of valid exponents .Comment: Revised according to referee's comments. 22 pages, to appear in J.
Funct. Ana
On the boundedness of the Wave Operators for fourth order Schr\"odinger operators
We consider the fourth order Schr\"odinger operator in
three dimensions with real-valued potential . Let , if
decays sufficiently and there are no eigenvalues or resonances in the
absolutely continuous spectrum of then the wave operators extend to bounded operators on
for all .Comment: 22 pages. Fixed typos and addressed comments from referee report. To
appear in the Trans. Amer. Math. So
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