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Travelling waves for an epidemic model with non-smooth treatment rates
This is the post-print version of the final published paper that is available from the link below. Copyright @ 2010 IOP Publishing Ltd and SISSA.We consider a susceptible–infected–removed (SIR) epidemic model with two types of nonlinear treatment rates: (i) piecewise linear treatment rate with saturation effect, (ii) piecewise constant treatment rate with a jump (Heaviside function). For case (i), we compute travelling front solutions whose profiles are heteroclinic orbits which connect either the disease-free state to an infective state or two endemic states with each other. For case (ii), it is shown that the profile has the following properties: the number of susceptibles is monotonically increasing and the number of infectives approaches zero at infinity, while their product converges to a constant. Numerical simulations are performed for all these cases. Abnormal behaviour like travelling waves with non-monotonic profile or oscillations is observed
Numerical Solution of Quantum-Mechanical Pair Equations
We discuss and illustrate the numerical solution of the differential equation satisfied by the first‐order pair functions of Sinanoğlu. An expansion of the pair function in spherical harmonics and the use of finite difference methods convert the differential equation into a set of simultaneous equations. Large systems of such equations can be solved economically. The method is simple and straightforward, and we have applied it to the first‐order pair function for helium with 1 / r_(12) as the perturbation. The results are accurate and encouraging, and since the method is numerical they are indicative of its potential for obtaining atomic‐pair functions in general
Subtree prune and regraft: a reversible real tree-valued Markov process
We use Dirichlet form methods to construct and analyze a reversible Markov
process, the stationary distribution of which is the Brownian continuum random
tree. This process is inspired by the subtree prune and regraft (SPR) Markov
chains that appear in phylogenetic analysis. A key technical ingredient in this
work is the use of a novel Gromov--Hausdorff type distance to metrize the space
whose elements are compact real trees equipped with a probability measure.
Also, the investigation of the Dirichlet form hinges on a new path
decomposition of the Brownian excursion.Comment: Published at http://dx.doi.org/10.1214/009117906000000034 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Terrestrial Planet Formation in a protoplanetary disk with a local mass depletion: A successful scenario for the formation of Mars
Models of terrestrial planet formation for our solar system have been
successful in producing planets with masses and orbits similar to those of
Venus and Earth. However, these models have generally failed to produce
Mars-sized objects around 1.5 AU. The body that is usually formed around Mars'
semimajor axis is, in general, much more massive than Mars. Only when Jupiter
and Saturn are assumed to have initially very eccentric orbits (e 0.1),
which seems fairly unlikely for the solar system, or alternately, if the
protoplanetary disk is truncated at 1.0 AU, simulations have been able to
produce Mars-like bodies in the correct location. In this paper, we examine an
alternative scenario for the formation of Mars in which a local depletion in
the density of the protosolar nebula results in a non-uniform formation of
planetary embryos and ultimately the formation of Mars-sized planets around 1.5
AU. We have carried out extensive numerical simulations of the formation of
terrestrial planets in such a disk for different scales of the local density
depletion, and for different orbital configurations of the giant planets. Our
simulations point to the possibility of the formation of Mars-sized bodies
around 1.5 AU, specifically when the scale of the disk local mass-depletion is
moderately high (50-75%) and Jupiter and Saturn are initially in their current
orbits. In these systems, Mars-analogs are formed from the protoplanetary
materials that originate in the regions of disk interior or exterior to the
local mass-depletion. Results also indicate that Earth-sized planets can form
around 1 AU with a substantial amount of water accreted via primitive
water-rich planetesimals and planetary embryos. We present the results of our
study and discuss their implications for the formation of terrestrial planets
in our solar system.Comment: Accepted for publication in The Astrophysical Journa
More Schooling, More Children: Compulsory Schooling Reforms and Fertility in Europe
We study the relationship between education and fertility, exploiting compulsory schooling reforms in Europe as source of exogenous variation in education. Using data from 8 European countries, we assess the causal effect of education on the number of biological kids and the incidence of childlessness. We find that more education causes a substantial decrease in childlessness and an increase in the average number of children per woman. Our findings are robust to a number of falsification checks and we can provide complementary empirical evidence on the mechanisms leading to these surprising results.
Terrestrial Planet Formation Constrained by Mars and the Structure of the Asteroid Belt
Reproducing the large Earth/Mars mass ratio requires a strong mass depletion
in solids within the protoplanetary disk between 1 and 3 AU. The Grand Tack
model invokes a specific migration history of the giant planets to remove most
of the mass initially beyond 1 AU and to dynamically excite the asteroid belt.
However, one could also invoke a steep density gradient created by inward drift
and pile-up of small particles induced by gas-drag, as has been proposed to
explain the formation of close-in super Earths. Here we show that the asteroid
belt's orbital excitation provides a crucial constraint against this scenario
for the Solar System. We performed a series of simulations of terrestrial
planet formation and asteroid belt evolution starting from disks of
planetesimals and planetary embryos with various radial density gradients and
including Jupiter and Saturn on nearly circular and coplanar orbits. Disks with
shallow density gradients reproduce the dynamical excitation of the asteroid
belt by gravitational self-stirring but form Mars analogs significantly more
massive than the real planet. In contrast, a disk with a surface density
gradient proportional to reproduces the Earth/Mars mass ratio but
leaves the asteroid belt in a dynamical state that is far colder than the real
belt. We conclude that no disk profile can simultaneously explain the structure
of the terrestrial planets and asteroid belt. The asteroid belt must have been
depleted and dynamically excited by a different mechanism such as, for
instance, in the Grand Tack scenario.Comment: Accepted for publication in MNRA
Localization and its consequences for quantum walk algorithms and quantum communication
The exponential speed-up of quantum walks on certain graphs, relative to
classical particles diffusing on the same graph, is a striking observation. It
has suggested the possibility of new fast quantum algorithms. We point out here
that quantum mechanics can also lead, through the phenomenon of localization,
to exponential suppression of motion on these graphs (even in the absence of
decoherence). In fact, for physical embodiments of graphs, this will be the
generic behaviour. It also has implications for proposals for using spin
networks, including spin chains, as quantum communication channels.Comment: 4 pages, 1 eps figure. Updated references and cosmetic changes for v
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