4,727 research outputs found
Generating upward sweeps in population using the Turchin--Korotayev model
The works of [Cha-DunAlvInoNieCarFieLaw,Cha-Dun] describe upward sweeps in
populations of city-states and attempt to characterize such phenomenon. The
model proposed in both [TurKor,Tur] describes how the population, state
resources and internal conflict influence each other over time. We show that
one can obtain an upward sweep in the population by altering particular
parameters of the system of differential equations constituting the model given
in [TurKor,Tur]. Moreover, we show that such a system has a unstable critical
point and propose an approach for determining bifurcation points in the
parameter space for the model.Comment: 20 pages, 13 figures. Contains Matlab code for those interested in
adapting and/or extending the model. Comments are welcom
Properties of the flow on a polygonal Andreev billiard
A formal definition of a (mathematical) polygonal Andreev billiard and a
construction of an equivalence relation that captures the dynamics described in
physical toy model of Andreev reflection are given. The continuous flow and
discrete flow on the respective phase spaces. It is then shown that the
continuous flow preserves the absolute value of the volume element and the billiard (collision) map preserves the measure , respectively. One can then characterize the dynamics of a
rational polygonal Andreev billiard table. Finally, a discussions of the effect
of a fractal perturbation of the toy model of a rectangular nanowire lying upon
a superconducting medium is given.Comment: This is a preliminary report and open for comments and suggestions.
17 pages, 15 figures. Makes reference to arXiv:1503.0849
Sequences of compatible periodic hybrid orbits of prefractal Koch snowflake billiards
The Koch snowflake KS is a nowhere differentiable curve. The billiard table
Omega(KS) with boundary KS is, a priori, not well defined. That is, one cannot
a priori determine the minimal path traversed by a billiard ball subject to a
collision in the boundary of the table. It is this problem which makes
Omega(KS) such an interesting, yet difficult, table to analyze. In this paper,
we approach this problem by approximating (from the inside) Omega(KS) by
well-defined (prefractal) rational polygonal billiard tables Omega(KS_n). We
first show that the flat surface S(KS_n) determined from the rational billiard
Omega(KS_n) is a branched cover of the singly punctured hexagonal torus. Such a
result, when combined with the results of [Gut2], allows us to define a
sequence of compatible orbits of prefractal billiards. We define a hybrid orbit
of a prefractal billiard Omega(KS_n) and show that every dense orbit of a
prefractal billiard is a dense hybrid orbit of Omega(KS_n). This result is key
in obtaining a topological dichotomy for a sequence of compatible orbits.
Furthermore, we determine a sufficient condition for a sequence of compatible
orbits to be a sequence of compatible periodic hybrid orbits. We then examine
the limiting behavior of a sequence of compatible periodic hybrid orbits. We
show that the trivial limit of particular (eventually) constant sequences of
compatible hybrid orbits constitutes an orbit of Omega(KS). In addition, we
show that the union of two suitably chosen nontrivial polygonal paths connects
two elusive limit points of the Koch snowflake. Finally, we discuss how it may
be possible for our results to be generalized to other fractal billiard tables
and how understanding the structures of the Veech groups of the prefractal
billiards may help in determining `fractal flat surfaces' naturally associated
with the billiard flows.Comment: 21 pages, 15 figure
The Wild, Elusive Singularities of the T-fractal Surface
We give a rigorous definition of the T-fractal translation surface, and
describe some its basic geometric and dynamical properties. In particular, we
study the singularities attached to the surface by its metric completion and
show there exists a Cantor set of "elusive singularities." We show these
elusive singularities can be thought of as a generalization of the wild
singularities introduced by Bowman and Valdez. In particular, we show that
every elusive singularities has an infinite discrete set of rotational
components.Comment: 30 pages, 18 figures. Comments welcom
Families of Periodic Orbits of the Koch Snowflake Fractal Billiard
We describe the periodic orbits of the prefractal Koch snowflake billiard
(the nth inner rational polygonal approximation of the Koch snowflake
billiard). In the case of the finite (prefractal) billiard table, we focus on
the direction given by an initial angle of pi/3, and define 1) a compatible
sequence of piecewise Fagnano orbits, 2) an eventually constant compatible
sequence of orbits and 3) a compatible sequence of generalized piecewise
Fagnano orbits. In the case of the infinite (fractal) billiard table, we will
describe what we call stabilizing periodic orbits of the Koch snowflake fractal
billiard. In a sense, we show that it is possible to define billiard dynamics
on a Cantor set. In addition, we will show that the inverse limit of the
footprints of orbits of the prefractal approximations exists in a specific
situation and provide a plausibility argument as to why such an inverse limit
of footprints should constitute the footprint of a well-defined periodic orbit
of the fractal billiard. Using known results for the inverse limit of a
sequence of finite spaces, we deduce that the footprint (i.e., the intersection
of the orbit with the boundary) of a piecewise Fagnano orbit is a topological
Cantor set and a self-similar Cantor set. We allude to a possible
characterization of orbits with an initial direction of pi/3. Such a
characterization would allow one to describe an orbit with an initial direction
of pi/3 of the Koch snowflake billiard as either a piecewise Fagnano orbit, a
stabilizing orbit or a generalized piecewise Fagnano orbit. We then close the
paper by discussing several outstanding open problems and conjectures about the
Koch snowflake fractal billiard, the associated 'fractal flat surface', and
possible connections with the associated fractal drum. In the long-term, the
present work may help lay the foundations for a general theory of fractal
billiards.Comment: This paper contains an index of notation, a table of contents and 31
figures; it is 63 pages in lengt
Towards the Koch Snowflake Fractal Billiard: Computer Experiments and Mathematical Conjectures
In this paper, we attempt to define and understand the orbits of the Koch
snowflake fractal billiard . This is a priori a very difficult problem
because , the snowflake curve boundary of , is nowhere
differentiable, making it impossible to apply the usual law of reflection at
any point of the boundary of the billiard table. Consequently, we view the
prefractal billiards (naturally approximating from the inside) as
rational polygonal billiards and examine the corresponding flat surfaces of
, denoted by . In order to develop a clearer picture
of what may possibly be happening on the billiard , we simulate billiard
trajectories on (at first, for a fixed ). Such computer
experiments provide us with a wealth of questions and lead us to formulate
conjectures about the existence and the geometric properties of periodic orbits
of and detail a possible plan on how to prove such conjectures.Comment: 26 pages, color figures (For crisper figures, please contact the
second author
Nontrivial paths and periodic orbits of the -fractal billiard table
We introduce and prove numerous new results about the orbits of the
-fractal billiard. Specifically, in Section 3, we give a variety of
sufficient conditions for the existence of a sequence of compatible periodic
orbits. In Section 4, we examine the limiting behavior of particular sequences
of compatible periodic orbits and, more interesting, in Section 5, the limiting
behavior of a particular sequence of compatible singular orbits. The latter
seems to indicate that the classification of orbits may not be so
straightforward. Additionally, sufficient conditions for the existence of
particular nontrivial paths is given in Section 4. The proofs of two results
stated in [LapNie4] appear here for the first time, as well. A discussion of
our results and directions for future research is then given in Section 6.Comment: 20 Figures, 35 pages, two results from arXiv:1210.0282 are
generalized and proved in this article. Many new results appear here.
Comments welcome. Version 3 contains minor grammatical changes and the
presentation of some results has greatly improved. To appear in the journal
Nonlinearit
Impact of baryonic streaming velocities on the formation of supermassive black holes via direct collapse
Baryonic streaming motions produced prior to the epoch of recombination
became supersonic during the cosmic dark ages. Various studies suggest that
such streaming velocities change the halo statistics and also influence the
formation of Population III stars. In this study, we aim to explore the impact
of streaming velocities on the formation of supermassive black holes at
via the direct collapse scenario. To accomplish this goal, we perform
cosmological large eddy simulations for two halos of a few times with initial streaming velocities of 3, 6 and 9 . These
massive primordial halos illuminated by the strong Lyman Werner flux are the
potential cradles for the formation of direct collapse seed black holes. To
study the evolution for longer times, we employ sink particles and track the
accretion for 10,000 years. Our findings show that higher streaming velocities
increase the circular velocities from about 14 to 16 .
They also delay the collapse of halos for a few million years, but do not have
any significant impact on the halo properties such as turbulent energy, radial
velocity, density and accretion rates. Sink particles of about are formed at the end of our simulations and no clear distribution
of sink masses is observed in the presence of streaming motions. It is further
found that the impact of streaming velocities is less severe in massive halos
compared to the minihalos as reported in the previous studies.Comment: Matches the accepted vesion, to be appeared MNRA
The formation of massive Pop III stars in the presence of turbulence
Population III stars forming in the infant universe at z=30 heralded the end
of the cosmic dark ages. They are presumed to be assembled in so-called
minihaloes with virial temperatures of a few thousand K where collapse is
triggered by molecular hydrogen cooling. A central question concerns their
final masses, and whether fragmentation occurs during their formation. While
studies employing Lagrangian codes suggest fragmentation via a self-gravitating
disk, recent high resolution simulations indicated that disk formation is
suppressed. Here we report the first high-resolution large-eddy simulations
performed with the Eulerian grid-based code Enzo following the evolution beyond
the formation of the first peak, to investigate the accretion of the central
massive clump and potential fragmentation. For a total of 3 halos, we see that
a disk forms around the first clump. The central clump reaches solar
masses after 40 years, while subsequent accretion is expected at a rate of
solar masses per year. In one of these halos, additional clumps form
as a result of fragmentation which proceeds at larger scales. We note that
subgrid-scale turbulence yields relevant contributions to the stability of the
protostellar disks. We conclude that the first protostar may reach masses up to
, which are only limited by the effect of radiative
feedback.Comment: Accepted for publication in APJL, comments are welcom
The small scale dynamo and the amplification of magnetic fields in massive primordial haloes
While present standard model of cosmology yields no clear prediction for the
initial magnetic field strength, efficient dynamo action may compensate for
initially weak seed fields via rapid amplification. In particular, the
small-scale dynamo is expected to exponentially amplify any weak magnetic field
in the presence of turbulence. We explore whether this scenario is viable using
cosmological magneto-hydrodynamics simulations modeling the formation of the
first galaxies, which are expected to form in so-called atomic cooling halos
with virial temperatures K. As previous calculations
have shown that a high Jeans resolution is needed to resolve turbulent
structures and dynamo effects, our calculations employ resolutions of up to 128
cells per Jeans length. The presence of the dynamo can be clearly confirmed for
resolutions of at least 64 cells per Jeans length, while saturation occurs at
approximate equipartition with turbulent energy. As a result of the large
Reynolds numbers in primordial galaxies, we expect saturation to occur at early
stages, implying magnetic field strengths of \sim0.1 G at densities of
10^4 cm^{-3}.Comment: Matches the accepted version to be appeared in MNRA
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