104 research outputs found
Stability aspects of the traveling salesman problem based on k-best solutions
AbstractThis paper discusses stability analysis for the Traveling Salesman Problem (TSP). For a traveling salesman tour which is known to be optimal with respect to a given instance (length vector) we are interested in determining the stability region, i.e. the set of all length vectors for which the tour is optimal. The following three subsets of the stability region are of special interest: 1.(1) tolerances, i.e. the maximum perturbations of single edges;2.(2) tolerance regions which are subsets of the stability region that can be constructed from the tolerances; and3.(3) the largest ball contained in the stability region centered at the given length vector (the corresponding radius is known as the stability radius).
It is well known that the problems of determining tolerances and the stability radius for the TSP are NP-hard so that in general it is not possible to obtain the above-mentioned three subsets without spending a lot of computation time. The question addressed in this paper is the following: assume that not only an optimal tour is known, but also a set of k shortest tours (k ⩾2) is given. Then to which extent does this allow us to determine the three subsets in polynomial time? It will be shown in this paper that having k-best solutions can give the desired information only partially. More precisely, it will be shown that only some of the tolerances can be determined exactly and for the other ones as well as for the stability radius only lower and/or upper bounds can be derived. Since the amount of information that can be derived from the set of k-best solutions is dependent on both the value of k as well as on the specific length vector, we present numerical experiments on instances from the TSPLIB library to analyze the effectiveness of our approach
Von Neumann equations with time-dependent Hamiltonians and supersymmetric quantum mechanics
Starting with a time-independent Hamiltonian and an appropriately chosen
solution of the von Neumann equation we construct
its binary-Darboux partner and an exact scattering solution of
where is time-dependent and not
isospectral to . The method is analogous to supersymmetric quantum mechanics
but is based on a different version of a Darboux transformation. We illustrate
the technique by the example where corresponds to a 1-D harmonic
oscillator. The resulting represents a scattering of a soliton-like
pulse on a three-level system.Comment: revtex, 3 eps file
Dynamical System Approach to Cosmological Models with a Varying Speed of Light
Methods of dynamical systems have been used to study homogeneous and
isotropic cosmological models with a varying speed of light (VSL). We propose
two methods of reduction of dynamics to the form of planar Hamiltonian
dynamical systems for models with a time dependent equation of state. The
solutions are analyzed on two-dimensional phase space in the variables where is a function of a scale factor . Then we show how the
horizon problem may be solved on some evolutional paths. It is shown that the
models with negative curvature overcome the horizon and flatness problems. The
presented method of reduction can be adopted to the analysis of dynamics of the
universe with the general form of the equation of state .
This is demonstrated using as an example the dynamics of VSL models filled with
a non-interacting fluid. We demonstrate a new type of evolution near the
initial singularity caused by a varying speed of light. The singularity-free
oscillating universes are also admitted for positive cosmological constant. We
consider a quantum VSL FRW closed model with radiation and show that the
highest tunnelling rate occurs for a constant velocity of light if and . It is also proved that the considered class of
models is structurally unstable for the case of .Comment: 18 pages, 5 figures, RevTeX4; final version to appear in PR
Observational hints on the Big Bounce
In this paper we study possible observational consequences of the bouncing
cosmology. We consider a model where a phase of inflation is preceded by a
cosmic bounce. While we consider in this paper only that the bounce is due to
loop quantum gravity, most of the results presented here can be applied for
different bouncing cosmologies. We concentrate on the scenario where the scalar
field, as the result of contraction of the universe, is driven from the bottom
of the potential well. The field is amplified, and finally the phase of the
standard slow-roll inflation is realized. Such an evolution modifies the
standard inflationary spectrum of perturbations by the additional oscillations
and damping on the large scales. We extract the parameters of the model from
the observations of the cosmic microwave background radiation. In particular,
the value of inflaton mass is equal to GeV. In
our considerations we base on the seven years of observations made by the WMAP
satellite. We propose the new observational consistency check for the phase of
slow-roll inflation. We investigate the conditions which have to be fulfilled
to make the observations of the Big Bounce effects possible. We translate them
to the requirements on the parameters of the model and then put the
observational constraints on the model. Based on assumption usually made in
loop quantum cosmology, the Barbero-Immirzi parameter was shown to be
constrained by from the cosmological observations. We have
compared the Big Bounce model with the standard Big Bang scenario and showed
that the present observational data is not informative enough to distinguish
these models.Comment: 25 pages, 8 figures, JHEP3.cl
Uniting cosmological epochs through the twister solution in cosmology with non-minimal coupling
We investigate dynamics of a flat FRW cosmological model with a barotropic
matter and a non-minimally coupled scalar field (both canonical and phantom).
In our approach we do not assume any specific form of a potential function for
the scalar field and we are looking for generic scenarios of evolution. We show
that dynamics of universe can be reduced to a 3-dimensional dynamical system.
We have found the set of fixed points and established their character. These
critical points represent all important epochs in evolution of the universe :
(a) a finite scale factor singularity, (b) an inflation (rapid-roll and
slow-roll), (c) a radiation domination, (d) a matter domination and (e) a
quintessence era. We have shown that the inflation, the radiation and matter
domination epochs are transient ones and last for a finite amount of time. The
existence of the radiation domination epoch is purely the effect of a
non-minimal coupling constant. We show the existence of a twister type solution
wandering between all these critical points.Comment: 22 pages, 5 figs; (v2.) 27 pages, 12 figs, JCAP in pres
Particle-Like Description in Quintessential Cosmology
Assuming equation of state for quintessential matter: , we
analyse dynamical behaviour of the scale factor in FRW cosmologies. It is shown
that its dynamics is formally equivalent to that of a classical particle under
the action of 1D potential . It is shown that Hamiltonian method can be
easily implemented to obtain a classification of all cosmological solutions in
the phase space as well as in the configurational space. Examples taken from
modern cosmology illustrate the effectiveness of the presented approach.
Advantages of representing dynamics as a 1D Hamiltonian flow, in the analysis
of acceleration and horizon problems, are presented. The inverse problem of
reconstructing the Hamiltonian dynamics (i.e. potential function) from the
luminosity distance function for supernovae is also considered.Comment: 35 pages, 26 figures, RevTeX4, some applications of our treatment to
investigation of quintessence models were adde
Scalar field cosmology in the energy phase-space -- unified description of dynamics
In this letter we apply dynamical system methods to study all evolutional
paths admissible for all initial conditions of the FRW cosmological model with
a non-minimally coupled to gravity scalar field and a barotropic fluid. We
choose "energy variables" as phase variables. We reduce dynamics to a
3-dimensional dynamical system for an arbitrary potential of the scalar field
in the phase space variables. After postulating the potential parameter
as a function of (defined as ) we reduce whole
dynamics to a 3-dimensional dynamical system and study evolutional paths
leading to current accelerating expansion. If we restrict the form of the
potential then we will obtain a 2-dimensional dynamical system. We use the
dynamical system approach to find a new generic quintessence scenario of
approaching to the de Sitter attractor which appears only for the case of
non-vanishing coupling constant.Comment: revtex4, 16 pages, 3 figs; (v2) refs. added, published versio
On the absence of ferromagnetism in typical 2D ferromagnets
We consider the Ising systems in dimensions with nearest-neighbor
ferromagnetic interactions and long-range repulsive (antiferromagnetic)
interactions which decay with a power, , of the distance. The physical
context of such models is discussed; primarily this is and where,
at long distances, genuine magnetic interactions between genuine magnetic
dipoles are of this form. We prove that when the power of decay lies above
and does not exceed , then for all temperatures, the spontaneous
magnetization is zero. In contrast, we also show that for powers exceeding
(with ) magnetic order can occur.Comment: 15 pages, CMP style fil
Simple Dynamics on the Brane
We apply methods of dynamical systems to study the behaviour of the
Randall-Sundrum models. We determine evolutionary paths for all possible
initial conditions in a 2-dimensional phase space and we investigate the set of
accelerated models. The simplicity of our formulation in comparison to some
earlier studies is expressed in the following: our dynamical system is a
2-dimensional Hamiltonian system, and what is more advantageous, it is free
from the degeneracy of critical points so that the system is structurally
stable. The phase plane analysis of Randall-Sundrum models with isotropic
Friedmann geometry clearly shows that qualitatively we deal with the same types
of evolution as in general relativity, although quantitatively there are
important differences.Comment: an improved version, 34 pages, 9 eps figure
Estimating the Fractal Dimension, K_2-entropy, and the Predictability of the Atmosphere
The series of mean daily temperature of air recorded over a period of 215
years is used for analysing the dimensionality and the predictability of the
atmospheric system. The total number of data points of the series is 78527.
Other 37 versions of the original series are generated, including ``seasonally
adjusted'' data, a smoothed series, series without annual course, etc. Modified
methods of Grassberger and Procaccia are applied. A procedure for selection of
the ``meaningful'' scaling region is proposed. Several scaling regions are
revealed in the ln C(r) versus ln r diagram. The first one in the range of
larger ln r has a gradual slope and the second one in the range of intermediate
ln r has a fast slope. Other two regions are settled in the range of small ln
r. The results lead us to claim that the series arises from the activity of at
least two subsystems. The first subsystem is low-dimensional (d_f=1.6) and it
possesses the potential predictability of several weeks. We suggest that this
subsystem is connected with seasonal variability of weather. The second
subsystem is high-dimensional (d_f>17) and its error-doubling time is about 4-7
days. It is found that the predictability differs in dependence on season. The
predictability time for summer, winter and the entire year (T_2 approx. 4.7
days) is longer than for transition-seasons (T_2 approx. 4.0 days for spring,
T_2 approx. 3.6 days for autumn). The role of random noise and the number of
data points are discussed. It is shown that a 15-year-long daily temperature
series is not sufficient for reliable estimations based on Grassberger and
Procaccia algorithms.Comment: 27 pages (LaTex version 2.09) and 15 figures as .ps files, e-mail:
[email protected]
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