655,863 research outputs found
Sharp entrywise perturbation bounds for Markov chains
For many Markov chains of practical interest, the invariant distribution is
extremely sensitive to perturbations of some entries of the transition matrix,
but insensitive to others; we give an example of such a chain, motivated by a
problem in computational statistical physics. We have derived perturbation
bounds on the relative error of the invariant distribution that reveal these
variations in sensitivity.
Our bounds are sharp, we do not impose any structural assumptions on the
transition matrix or on the perturbation, and computing the bounds has the same
complexity as computing the invariant distribution or computing other bounds in
the literature. Moreover, our bounds have a simple interpretation in terms of
hitting times, which can be used to draw intuitive but rigorous conclusions
about the sensitivity of a chain to various types of perturbations
A simple encoding of a quantum circuit amplitude as a matrix permanent
A simple construction is presented which allows computing the transition
amplitude of a quantum circuit to be encoded as computing the permanent of a
matrix which is of size proportional to the number of quantum gates in the
circuit. This opens up some interesting classical monte-carlo algorithms for
approximating quantum circuits.Comment: 6 figure
Transition Temperature of Dilute, Weakly Repulsive Bose Gas
Within a quasiparticle framework, we reconsider the issue of computing the
Bose-Einstein condensation temperature () in a weakly non-ideal Bose gas.
The main result of this and previous investigations is that increases
with the scattering length , with the leading dependence being either linear
or log-linear in . The calculation of reduces to that of computing the
excitation spectrum near the transition. We report two approaches to
regularizing the infrared divergence at the transition point. One leads to a
-like shift in , and the other allows numerical
calculations for the shift.Comment: 8 pages, 3 figures, revtex
Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution
A birth-death process is a continuous-time Markov chain that counts the
number of particles in a system over time. In the general process with
current particles, a new particle is born with instantaneous rate
and a particle dies with instantaneous rate . Currently no robust and
efficient method exists to evaluate the finite-time transition probabilities in
a general birth-death process with arbitrary birth and death rates. In this
paper, we first revisit the theory of continued fractions to obtain expressions
for the Laplace transforms of these transition probabilities and make explicit
an important derivation connecting transition probabilities and continued
fractions. We then develop an efficient algorithm for computing these
probabilities that analyzes the error associated with approximations in the
method. We demonstrate that this error-controlled method agrees with known
solutions and outperforms previous approaches to computing these probabilities.
Finally, we apply our novel method to several important problems in ecology,
evolution, and genetics
Two-qubit Quantum Logic Gate in Molecular Magnets
We proposed a scheme to realize a controlled-NOT quantum logic gate in a
dimer of exchange coupled single-molecule magnets, . We
chosen the ground state and the three low-lying excited states of a dimer in a
finite longitudinal magnetic field as the quantum computing bases and
introduced a pulsed transverse magnetic field with a special frequency. The
pulsed transverse magnetic field induces the transitions between the quantum
computing bases so as to realize a controlled-NOT quantum logic gate. The
transition rates between the quantum computing bases and between the quantum
computing bases and other excited states are evaluated and analyzed.Comment: 7 pages, 2 figure
String Method for the Study of Rare Events
We present a new and efficient method for computing the transition pathways,
free energy barriers, and transition rates in complex systems with relatively
smooth energy landscapes. The method proceeds by evolving strings, i.e. smooth
curves with intrinsic parametrization whose dynamics takes them to the most
probable transition path between two metastable regions in the configuration
space. Free energy barriers and transition rates can then be determined by
standard umbrella sampling technique around the string. Applications to
Lennard-Jones cluster rearrangement and thermally induced switching of a
magnetic film are presented.Comment: 4 pages, 4 figure
Efficient Transition Probability Computation for Continuous-Time Branching Processes via Compressed Sensing
Branching processes are a class of continuous-time Markov chains (CTMCs) with
ubiquitous applications. A general difficulty in statistical inference under
partially observed CTMC models arises in computing transition probabilities
when the discrete state space is large or uncountable. Classical methods such
as matrix exponentiation are infeasible for large or countably infinite state
spaces, and sampling-based alternatives are computationally intensive,
requiring a large integration step to impute over all possible hidden events.
Recent work has successfully applied generating function techniques to
computing transition probabilities for linear multitype branching processes.
While these techniques often require significantly fewer computations than
matrix exponentiation, they also become prohibitive in applications with large
populations. We propose a compressed sensing framework that significantly
accelerates the generating function method, decreasing computational cost up to
a logarithmic factor by only assuming the probability mass of transitions is
sparse. We demonstrate accurate and efficient transition probability
computations in branching process models for hematopoiesis and transposable
element evolution.Comment: 18 pages, 4 figures, 2 table
Retraction and Generalized Extension of Computing with Words
Fuzzy automata, whose input alphabet is a set of numbers or symbols, are a
formal model of computing with values. Motivated by Zadeh's paradigm of
computing with words rather than numbers, Ying proposed a kind of fuzzy
automata, whose input alphabet consists of all fuzzy subsets of a set of
symbols, as a formal model of computing with all words. In this paper, we
introduce a somewhat general formal model of computing with (some special)
words. The new features of the model are that the input alphabet only comprises
some (not necessarily all) fuzzy subsets of a set of symbols and the fuzzy
transition function can be specified arbitrarily. By employing the methodology
of fuzzy control, we establish a retraction principle from computing with words
to computing with values for handling crisp inputs and a generalized extension
principle from computing with words to computing with all words for handling
fuzzy inputs. These principles show that computing with values and computing
with all words can be respectively implemented by computing with words. Some
algebraic properties of retractions and generalized extensions are addressed as
well.Comment: 13 double column pages; 3 figures; to be published in the IEEE
Transactions on Fuzzy System
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