118 research outputs found
On the efficiency of Hamiltonian-based quantum computation for low-rank matrices
We present an extension of Adiabatic Quantum Computing (AQC) algorithm for
the unstructured search to the case when the number of marked items is unknown.
The algorithm maintains the optimal Grover speedup and includes a small
counting subroutine.
Our other results include a lower bound on the amount of time needed to
perform a general Hamiltonian-based quantum search, a lower bound on the
evolution time needed to perform a search that is valid in the presence of
control error and a generic upper bound on the minimum eigenvalue gap for
evolutions.
In particular, we demonstrate that quantum speedup for the unstructured
search using AQC type algorithms may only be achieved under very rigid control
precision requirements.Comment: 17 pages, no figures, to appear in JM
Connecting protein and mRNA burst distributions for stochastic models of gene expression
The intrinsic stochasticity of gene expression can lead to large variability
in protein levels for genetically identical cells. Such variability in protein
levels can arise from infrequent synthesis of mRNAs which in turn give rise to
bursts of protein expression. Protein expression occurring in bursts has indeed
been observed experimentally and recent studies have also found evidence for
transcriptional bursting, i.e. production of mRNAs in bursts. Given that there
are distinct experimental techniques for quantifying the noise at different
stages of gene expression, it is of interest to derive analytical results
connecting experimental observations at different levels. In this work, we
consider stochastic models of gene expression for which mRNA and protein
production occurs in independent bursts. For such models, we derive analytical
expressions connecting protein and mRNA burst distributions which show how the
functional form of the mRNA burst distribution can be inferred from the protein
burst distribution. Additionally, if gene expression is repressed such that
observed protein bursts arise only from single mRNAs, we show how observations
of protein burst distributions (repressed and unrepressed) can be used to
completely determine the mRNA burst distribution. Assuming independent
contributions from individual bursts, we derive analytical expressions
connecting means and variances for burst and steady-state protein
distributions. Finally, we validate our general analytical results by
considering a specific reaction scheme involving regulation of protein bursts
by small RNAs. For a range of parameters, we derive analytical expressions for
regulated protein distributions that are validated using stochastic
simulations. The analytical results obtained in this work can thus serve as
useful inputs for a broad range of studies focusing on stochasticity in gene
expression
Radiation Exposure and Mortality from Cardiovascular Disease and Cancer in Early NASA Astronauts: Space for Exploration
Of the many possible health challenges posed during extended exploratory missions to space, the effects of space radiation on cardiovascular disease and cancer are of particular concern. There are unique challenges to estimating those radiation risks; care and appropriate and rigorous methodology should be applied when considering small cohorts such as the NASA astronaut population. The objective of this work was to establish whether there is evidence for excess cardiovascular disease or cancer mortality in an early NASA astronaut cohort and determine if a correlation exists between space radiation exposure and mortality
Mean-Field- and Classical Limit of Many-Body Schr\"odinger Dynamics for Bosons
We present a new proof of the convergence of the N-particle Schroedinger
dynamics for bosons towards the dynamics generated by the Hartree equation in
the mean-field limit. For a restricted class of two-body interactions, we
obtain convergence estimates uniform in the Planck constant , up to an
exponentially small remainder. For h=0, the classical dynamics in the
mean-field limit is given by the Vlasov equation.Comment: Latex 2e, 18 page
Uniqueness and Nondegeneracy of Ground States for in
We prove uniqueness of ground state solutions for the
nonlinear equation in , where
and for and for . Here denotes the fractional Laplacian
in one dimension. In particular, we generalize (by completely different
techniques) the specific uniqueness result obtained by Amick and Toland for
and in [Acta Math., \textbf{167} (1991), 107--126]. As a
technical key result in this paper, we show that the associated linearized
operator is nondegenerate;
i.\,e., its kernel satisfies .
This result about proves a spectral assumption, which plays a central
role for the stability of solitary waves and blowup analysis for nonlinear
dispersive PDEs with fractional Laplacians, such as the generalized
Benjamin-Ono (BO) and Benjamin-Bona-Mahony (BBM) water wave equations.Comment: 45 page
Derivation of the cubic NLS and Gross-Pitaevskii hierarchy from manybody dynamics in based on spacetime norms
We derive the defocusing cubic Gross-Pitaevskii (GP) hierarchy in dimension
, from an -body Schr\"{o}dinger equation describing a gas of
interacting bosons in the GP scaling, in the limit . The
main result of this paper is the proof of convergence of the corresponding
BBGKY hierarchy to a GP hierarchy in the spaces introduced in our previous work
on the well-posedness of the Cauchy problem for GP hierarchies,
\cite{chpa2,chpa3,chpa4}, which are inspired by the solutions spaces based on
space-time norms introduced by Klainerman and Machedon in \cite{klma}. We note
that in , this has been a well-known open problem in the field. While our
results do not assume factorization of the solutions, consideration of
factorized solutions yields a new derivation of the cubic, defocusing nonlinear
Schr\"odinger equation (NLS) in .Comment: 44 pages, AMS Late
On the Rigorous Derivation of the 3D Cubic Nonlinear Schr\"odinger Equation with A Quadratic Trap
We consider the dynamics of the 3D N-body Schr\"{o}dinger equation in the
presence of a quadratic trap. We assume the pair interaction potential is
N^{3{\beta}-1}V(N^{{\beta}}x). We justify the mean-field approximation and
offer a rigorous derivation of the 3D cubic NLS with a quadratic trap. We
establish the space-time bound conjectured by Klainerman and Machedon [30] for
{\beta} in (0,2/7] by adapting and simplifying an argument in Chen and
Pavlovi\'c [7] which solves the problem for {\beta} in (0,1/4) in the absence
of a trap.Comment: Revised according to the referee report. Accepted to appear in
Archive for Rational Mechanics and Analysi
Understanding the Random Displacement Model: From Ground-State Properties to Localization
We give a detailed survey of results obtained in the most recent half decade
which led to a deeper understanding of the random displacement model, a model
of a random Schr\"odinger operator which describes the quantum mechanics of an
electron in a structurally disordered medium. These results started by
identifying configurations which characterize minimal energy, then led to
Lifshitz tail bounds on the integrated density of states as well as a Wegner
estimate near the spectral minimum, which ultimately resulted in a proof of
spectral and dynamical localization at low energy for the multi-dimensional
random displacement model.Comment: 31 pages, 7 figures, final version, to appear in Proceedings of
"Spectral Days 2010", Santiago, Chile, September 20-24, 201
Quantum Diffusion and Eigenfunction Delocalization in a Random Band Matrix Model
We consider Hermitian and symmetric random band matrices in
dimensions. The matrix elements , indexed by , are independent, uniformly distributed random variables if \abs{x-y}
is less than the band width , and zero otherwise. We prove that the time
evolution of a quantum particle subject to the Hamiltonian is diffusive on
time scales . We also show that the localization length of an
arbitrarily large majority of the eigenvectors is larger than a factor
times the band width. All results are uniform in the size
\abs{\Lambda} of the matrix.Comment: Minor corrections, Sections 4 and 11 update
Widths of the Hall Conductance Plateaus
We study the charge transport of the noninteracting electron gas in a
two-dimensional quantum Hall system with Anderson-type impurities at zero
temperature. We prove that there exist localized states of the bulk order in
the disordered-broadened Landau bands whose energies are smaller than a certain
value determined by the strength of the uniform magnetic field. We also prove
that, when the Fermi level lies in the localization regime, the Hall
conductance is quantized to the desired integer and shows the plateau of the
bulk order for varying the filling factor of the electrons rather than the
Fermi level.Comment: 94 pages, v2: a revision of Sec. 5; v3: an error in Sec. 7 is
corrected, major revisions of Sec. 7 and Appendix E, Sec. 7 is enlarged to
Secs. 7-12, minor corrections; v4: major revisions, accepted for publication
in Journal of Statistical Physics; v5: minor corrections, accepted versio
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