442 research outputs found
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
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
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
Fermi, Pasta, Ulam and a mysterious lady
It is reported that the numerical simulations of the Fermi-Pasta-Ulam problem
were performed by a young lady, Mary Tsingou. After 50 years of omission, it is
time for a proper recognition of her decisive contribution to the first ever
numerical experiment, central in the solitons and chaos theories, but also one
of the very first out-of-equilibrium statistical mechanics study. Let us quote
from now on the Fermi-Pasta-Ulam-Tsingou problem
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
Hierarchy of random deterministic chaotic maps with an invariant measure
Hierarchy of one and many-parameter families of random trigonometric chaotic
maps and one-parameter random elliptic chaotic maps of type with an
invariant measure have been introduced. Using the invariant measure
(Sinai-Ruelle-Bowen measure), the Kolmogrov-Sinai entropy of the random chaotic
maps have been calculated analytically, where the numerical simulations support
the resultsComment: 11 pages, Late
An early warning indicator for atmospheric blocking events using transfer operators
The existence of persistent midlatitude atmospheric flow regimes with
time-scales larger than 5-10 days and indications of preferred transitions
between them motivates to develop early warning indicators for such regime
transitions. In this paper, we use a hemispheric barotropic model together with
estimates of transfer operators on a reduced phase space to develop an early
warning indicator of the zonal to blocked flow transition in this model. It is
shown that, the spectrum of the transfer operators can be used to study the
slow dynamics of the flow as well as the non-Markovian character of the
reduction. The slowest motions are thereby found to have time scales of three
to six weeks and to be associated with meta-stable regimes (and their
transitions) which can be detected as almost-invariant sets of the transfer
operator. From the energy budget of the model, we are able to explain the
meta-stability of the regimes and the existence of preferred transition paths.
Even though the model is highly simplified, the skill of the early warning
indicator is promising, suggesting that the transfer operator approach can be
used in parallel to an operational deterministic model for stochastic
prediction or to assess forecast uncertainty
Velocity distributions in dissipative granular gases
Motivated by recent experiments reporting non-Gaussian velocity distributions
in driven dilute granular materials, we study by numerical simulation the
properties of 2D inelastic gases. We find theoretically that the form of the
observed velocity distribution is governed primarily by the coefficient of
restitution and , the ratio between the average number of
heatings and the average number of collisions in the gas. The differences in
distributions we find between uniform and boundary heating can then be
understood as different limits of , for and
respectively.Comment: 5 figure
Temperature dependence of thermal conductivity in 1D nonlinear lattices
We examine the temperature dependence of thermal conductivity of one
dimensional nonlinear (anharmonic) lattices with and without on-site potential.
It is found from computer simulation that the heat conductivity depends on
temperature via the strength of nonlinearity. Based on this correlation, we
make a conjecture in the effective phonon theory that the mean-free-path of the
effective phonon is inversely proportional to the strength of nonlinearity. We
demonstrate analytically and numerically that the temperature behavior of the
heat conductivity is not universal for 1D harmonic lattices
with a small nonlinear perturbation. The computer simulations of temperature
dependence of heat conductivity in general 1D nonlinear lattices are in good
agreements with our theoretic predictions. Possible experimental test is
discussed.Comment: 6 pages and 2 figures. Accepted for publication in Europhys. Let
Bethe Ansatz in the Bernoulli Matching Model of Random Sequence Alignment
For the Bernoulli Matching model of sequence alignment problem we apply the
Bethe ansatz technique via an exact mapping to the 5--vertex model on a square
lattice. Considering the terrace--like representation of the sequence alignment
problem, we reproduce by the Bethe ansatz the results for the averaged length
of the Longest Common Subsequence in Bernoulli approximation. In addition, we
compute the average number of nucleation centers of the terraces.Comment: 14 pages, 5 figures (some points are clarified
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