556 research outputs found
An exactly solvable model for driven dissipative systems
We introduce a solvable stochastic model inspired by granular gases for
driven dissipative systems. We characterize far from equilibrium steady states
of such systems through the non-Boltzmann energy distribution and compare
different measures of effective temperatures. As an example we demonstrate that
fluctuation-dissipation relations hold, however with an effective temperature
differing from the effective temperature defined from the average energy.Comment: Some further clarifications. No changes in results or conclusion
Lagrangian Based Methods for Coherent Structure Detection
There has been a proliferation in the development of Lagrangian analytical methods for detecting coherent structures in fluid flow transport, yielding a variety of qualitatively different approaches. We present a review of four approaches and demonstrate the utility of these methods via their application to the same sample analytic model, the canonical double-gyre flow, highlighting the pros and cons of each approach. Two of the methods, the geometric and probabilistic approaches, are well established and require velocity field data over the time interval of interest to identify particularly important material lines and surfaces, and influential regions, respectively. The other two approaches, implementing tools from cluster and braid theory, seek coherent structures based on limited trajectory data, attempting to partition the flow transport into distinct regions. All four of these approaches share the common trait that they are objective methods, meaning that their results do not depend on the frame of reference used. For each method, we also present a number of example applications ranging from blood flow and chemical reactions to ocean and atmospheric flows. (C) 2015 AIP Publishing LLC.ONR N000141210665Center for Nonlinear Dynamic
Hierarchy of Chaotic Maps with an Invariant Measure
We give hierarchy of one-parameter family F(a,x) of maps of the interval
[0,1] with an invariant measure. Using the measure, we calculate
Kolmogorov-Sinai entropy, or equivalently Lyapunov characteristic exponent, of
these maps analytically, where the results thus obtained have been approved
with numerical simulation. In contrary to the usual one-parameter family of
maps such as logistic and tent maps, these maps do not possess period doubling
or period-n-tupling cascade bifurcation to chaos, but they have single fixed
point attractor at certain parameter values, where they bifurcate directly to
chaos without having period-n-tupling scenario exactly at these values of
parameter whose Lyapunov characteristic exponent begins to be positive.Comment: 18 pages (Latex), 7 figure
Transport in time-dependent dynamical systems: Finite-time coherent sets
We study the transport properties of nonautonomous chaotic dynamical systems
over a finite time duration. We are particularly interested in those regions
that remain coherent and relatively non-dispersive over finite periods of time,
despite the chaotic nature of the system. We develop a novel probabilistic
methodology based upon transfer operators that automatically detects maximally
coherent sets. The approach is very simple to implement, requiring only
singular vector computations of a matrix of transitions induced by the
dynamics. We illustrate our new methodology on an idealized stratospheric flow
and in two and three dimensional analyses of European Centre for Medium Range
Weather Forecasting (ECMWF) reanalysis data
A note on Verhulst's logistic equation and related logistic maps
We consider the Verhulst logistic equation and a couple of forms of the
corresponding logistic maps. For the case of the logistic equation we show that
using the general Riccati solution only changes the initial conditions of the
equation. Next, we consider two forms of corresponding logistic maps reporting
the following results. For the map x_{n+1} = rx_n(1 - x_n) we propose a new way
to write the solution for r = -2 which allows better precision of the iterative
terms, while for the map x_{n+1}-x_n = rx_n(1 - x_{n+1}) we show that it
behaves identically to the logistic equation from the standpoint of the general
Riccati solution, which is also provided herein for any value of the parameter
r.Comment: 6 pages, 3 figures, 7 references with title
Method of constructing exactly solvable chaos
We present a new systematic method of constructing rational mappings as
ergordic transformations with nonuniform invariant measures on the unit
interval [0,1]. As a result, we obtain a two-parameter family of rational
mappings that have a special property in that their invariant measures can be
explicitly written in terms of algebraic functions of parameters and a
dynamical variable. Furthermore, it is shown here that this family is the most
generalized class of rational mappings possessing the property of exactly
solvable chaos on the unit interval, including the Ulam=Neumann map y=4x(1-x).
Based on the present method, we can produce a series of rational mappings
resembling the asymmetric shape of the experimentally obtained first return
maps of the Beloussof-Zhabotinski chemical reaction, and we can match some
rational functions with other experimentally obtained first return maps in a
systematic manner.Comment: 12 pages, 2 figures, REVTEX. Title was changed. Generalized Chebyshev
maps including the precise form of two-parameter generalized cubic maps were
added. Accepted for publication in Phys. Rev. E(1997
Size reconstructibility of graphs
The deck of a graph is given by the multiset of (unlabelled) subgraphs
. The subgraphs are referred to as the cards of .
Brown and Fenner recently showed that, for , the number of edges of a
graph can be computed from any deck missing 2 cards. We show that, for
sufficiently large , the number of edges can be computed from any deck
missing at most cards.Comment: 15 page
Longest increasing subsequence as expectation of a simple nonlinear stochastic PDE with a low noise intensity
We report some new observation concerning the statistics of Longest
Increasing Subsequences (LIS). We show that the expectation of LIS, its
variance, and apparently the full distribution function appears in statistical
analysis of some simple nonlinear stochastic partial differential equation
(SPDE) in the limit of very low noise intensity.Comment: 6 pages, 4 figures, reference adde
Deterministic and Probabilistic Binary Search in Graphs
We consider the following natural generalization of Binary Search: in a given
undirected, positively weighted graph, one vertex is a target. The algorithm's
task is to identify the target by adaptively querying vertices. In response to
querying a node , the algorithm learns either that is the target, or is
given an edge out of that lies on a shortest path from to the target.
We study this problem in a general noisy model in which each query
independently receives a correct answer with probability (a
known constant), and an (adversarial) incorrect one with probability .
Our main positive result is that when (i.e., all answers are
correct), queries are always sufficient. For general , we give an
(almost information-theoretically optimal) algorithm that uses, in expectation,
no more than queries, and identifies the target correctly with probability at
leas . Here, denotes the
entropy. The first bound is achieved by the algorithm that iteratively queries
a 1-median of the nodes not ruled out yet; the second bound by careful repeated
invocations of a multiplicative weights algorithm.
Even for , we show several hardness results for the problem of
determining whether a target can be found using queries. Our upper bound of
implies a quasipolynomial-time algorithm for undirected connected
graphs; we show that this is best-possible under the Strong Exponential Time
Hypothesis (SETH). Furthermore, for directed graphs, or for undirected graphs
with non-uniform node querying costs, the problem is PSPACE-complete. For a
semi-adaptive version, in which one may query nodes each in rounds, we
show membership in in the polynomial hierarchy, and hardness
for
Atom cooling by non-adiabatic expansion
Motivated by the recent discovery that a reflecting wall moving with a
square-root in time trajectory behaves as a universal stopper of classical
particles regardless of their initial velocities, we compare linear in time and
square-root in time expansions of a box to achieve efficient atom cooling. For
the quantum single-atom wavefunctions studied the square-root in time expansion
presents important advantages: asymptotically it leads to zero average energy
whereas any linear in time (constant box-wall velocity) expansion leaves a
non-zero residual energy, except in the limit of an infinitely slow expansion.
For finite final times and box lengths we set a number of bounds and cooling
principles which again confirm the superior performance of the square-root in
time expansion, even more clearly for increasing excitation of the initial
state. Breakdown of adiabaticity is generally fatal for cooling with the linear
expansion but not so with the square-root expansion.Comment: 4 pages, 4 figure
- …