306 research outputs found
Irredundant Triangular Decomposition
Triangular decomposition is a classic, widely used and well-developed way to
represent algebraic varieties with many applications. In particular, there
exist sharp degree bounds for a single triangular set in terms of intrinsic
data of the variety it represents, and powerful randomized algorithms for
computing triangular decompositions using Hensel lifting in the
zero-dimensional case and for irreducible varieties. However, in the general
case, most of the algorithms computing triangular decompositions produce
embedded components, which makes it impossible to directly apply the intrinsic
degree bounds. This, in turn, is an obstacle for efficiently applying Hensel
lifting due to the higher degrees of the output polynomials and the lower
probability of success. In this paper, we give an algorithm to compute an
irredundant triangular decomposition of an arbitrary algebraic set defined
by a set of polynomials in C[x_1, x_2, ..., x_n]. Using this irredundant
triangular decomposition, we were able to give intrinsic degree bounds for the
polynomials appearing in the triangular sets and apply Hensel lifting
techniques. Our decomposition algorithm is randomized, and we analyze the
probability of success
Some algebraic properties of differential operators
First, we study the subskewfield of rational pseudodifferential operators
over a differential field K generated in the skewfield of pseudodifferential
operators over K by the subalgebra of all differential operators.
Second, we show that the Dieudonne' determinant of a matrix
pseudodifferential operator with coefficients in a differential subring A of K
lies in the integral closure of A in K, and we give an example of a 2x2 matrix
differential operator with coefficients in A whose Dieudonne' determiant does
not lie in A.Comment: 15 page
A fitting formula for the merger timescale of galaxies in hierarchical clustering
We study galaxy mergers using a high-resolution cosmological hydro/N-body
simulation with star formation, and compare the measured merger timescales with
theoretical predictions based on the Chandrasekhar formula. In contrast to
Navarro et al., our numerical results indicate, that the commonly used equation
for the merger timescale given by Lacey and Cole, systematically underestimates
the merger timescales for minor mergers and overestimates those for major
mergers. This behavior is partly explained by the poor performance of their
expression for the Coulomb logarithm, \ln (m_pri/m_sat). The two alternative
forms \ln (1+m_pri/m_sat) and 1/2\ln [1+(m_pri/m_sat)^2] for the Coulomb
logarithm can account for the mass dependence of merger timescale successfully,
but both of them underestimate the merger time scale by a factor 2. Since \ln
(1+m_pri/m_sat) represents the mass dependence slightly better we adopt this
expression for the Coulomb logarithm. Furthermore, we find that the dependence
of the merger timescale on the circularity parameter \epsilon is much weaker
than the widely adopted power-law \epsilon^{0.78}, whereas
0.94*{\epsilon}^{0.60}+0.60 provides a good match to the data. Based on these
findings, we present an accurate and convenient fitting formula for the merger
timescale of galaxies in cold dark matter models.Comment: 16 pages, 14 figures, accepted for publication in ApJ, minor changes
in the last few sentences of the discussio
Gravitational detection of a low-mass dark satellite at cosmological distance
The mass-function of dwarf satellite galaxies that are observed around Local
Group galaxies substantially differs from simulations based on cold dark
matter: the simulations predict many more dwarf galaxies than are seen. The
Local Group, however, may be anomalous in this regard. A massive dark satellite
in an early-type lens galaxy at z = 0.222 was recently found using a new method
based on gravitational lensing, suggesting that the mass fraction contained in
substructure could be higher than is predicted from simulations. The lack of
very low mass detections, however, prohibited any constraint on their mass
function. Here we report the presence of a 1.9 +/- 0.1 x 10^8 M_sun dark
satellite in the Einstein-ring system JVAS B1938+666 at z = 0.881, where M_sun
denotes solar mass. This satellite galaxy has a mass similar to the Sagittarius
galaxy, which is a satellite of the Milky Way. We determine the logarithmic
slope of the mass function for substructure beyond the local Universe to be
alpha = 1.1^+0.6_-0.4, with an average mass-fraction of f = 3.3^+3.6_-1.8 %, by
combining data on both of these recently discovered galaxies. Our results are
consistent with the predictions from cold dark matter simulations at the 95 per
cent confidence level, and therefore agree with the view that galaxies formed
hierarchically in a Universe composed of cold dark matter.Comment: 25 pages, 7 figures, accepted for publication in Nature (19 January
2012
Mergers and Mass Accretion Rates in Galaxy Assembly: The Millennium Simulation Compared to Observations of z~2 Galaxies
Recent observations of UV-/optically selected, massive star forming galaxies
at z~2 indicate that the baryonic mass assembly and star formation history is
dominated by continuous rapid accretion of gas and internal secular evolution,
rather than by major mergers. We use the Millennium Simulation to build new
halo merger trees, and extract halo merger fractions and mass accretion rates.
We find that even for halos not undergoing major mergers the mass accretion
rates are plausibly sufficient to account for the high star formation rates
observed in z~2 disks. On the other hand, the fraction of major mergers in the
Millennium Simulation is sufficient to account for the number counts of
submillimeter galaxies (SMGs), in support of observational evidence that these
are major mergers. When following the fate of these two populations in the
Millennium Simulation to z=0, we find that subsequent mergers are not frequent
enough to convert all z~2 turbulent disks into elliptical galaxies at z=0.
Similarly, mergers cannot transform the compact SMGs/red sequence galaxies at
z~2 into observed massive cluster ellipticals at z=0. We argue therefore, that
secular and internal evolution must play an important role in the evolution of
a significant fraction of z~2 UV-/optically and submillimeter selected galaxy
populations.Comment: 5 pages, 4 figures, Accepted for publication in Ap
Red Galaxy Growth and the Halo Occupation Distribution
We have traced the past 7 Gyr of red galaxy stellar mass growth within dark
matter halos. We have determined the halo occupation distribution, which
describes how galaxies reside within dark matter halos, using the observed
luminosity function and clustering of 40,696 0.2<z<1.0 red galaxies in Bootes.
Half of 10^{11.9} Msun/h halos host a red central galaxy, and this fraction
increases with increasing halo mass. We do not observe any evolution of the
relationship between red galaxy stellar mass and host halo mass, although we
expect both galaxy stellar masses and halo masses to evolve over cosmic time.
We find that the stellar mass contained within the red population has doubled
since z=1, with the stellar mass within red satellite galaxies tripling over
this redshift range. In cluster mass halos most of the stellar mass resides
within satellite galaxies and the intra-cluster light, with a minority of the
stellar mass residing within central galaxies. The stellar masses of the most
luminous red central galaxies are proportional to halo mass to the power of a
third. We thus conclude that halo mergers do not always lead to rapid growth of
central galaxies. While very massive halos often double in mass over the past 7
Gyr, the stellar masses of their central galaxies typically grow by only 30%.Comment: Accepted for publication in the ApJ. 34 pages, 22 Figures, 5 Table
Shaping the Phase of a Single Photon
While the phase of a coherent light field can be precisely known, the phase
of the individual photons that create this field, considered individually,
cannot. Phase changes within single-photon wave packets, however, have
observable effects. In fact, actively controlling the phase of individual
photons has been identified as a powerful resource for quantum communication
protocols. Here we demonstrate the arbitrary phase control of a single photon.
The phase modulation is applied without affecting the photon's amplitude
profile and is verified via a two-photon quantum interference measurement,
which can result in the fermionic spatial behaviour of photon pairs. Combined
with previously demonstrated control of a single photon's amplitude, frequency,
and polarisation, the fully deterministic phase shaping presented here allows
for the complete control of single-photon wave packets.Comment: 4 pages, 4 figure
Zero Order Estimates for Analytic Functions
The primary goal of this paper is to provide a general multiplicity estimate.
Our main theorem allows to reduce a proof of multiplicity lemma to the study of
ideals stable under some appropriate transformation of a polynomial ring. In
particular, this result leads to a new link between the theory of polarized
algebraic dynamical systems and transcendental number theory. On the other
hand, it allows to establish an improvement of Nesterenko's conditional result
on solutions of systems of differential equations. We also deduce, under some
condition on stable varieties, the optimal multiplicity estimate in the case of
generalized Mahler's functional equations, previously studied by Mahler,
Nishioka, Topfer and others. Further, analyzing stable ideals we prove the
unconditional optimal result in the case of linear functional systems of
generalized Mahler's type. The latter result generalizes a famous theorem of
Nishioka (1986) previously conjectured by Mahler (1969), and simultaneously it
gives a counterpart in the case of functional systems for an important
unconditional result of Nesterenko (1977) concerning linear differential
systems. In summary, we provide a new universal tool for transcendental number
theory, applicable with fields of any characteristic. It opens the way to new
results on algebraic independence, as shown in Zorin (2010).Comment: 42 page
A 700 year-old Pulsar in the Supernova Remnant Kes 75
Since their discovery 30 years ago, pulsars have been understood to be
neutron stars (NSs) born rotating rapidly (~ 10-100 ms). These neutron stars
are thought to be created in supernova explosions involving massive stars,
which give rise to expanding supernova remnants (SNRs). With over 220 Galactic
SNRs known (Green 1998) and over 1200 radio pulsars detected (Camilo et al.
2000), it is quite surprising that few associations between the two populations
have been identified with any certainty. Here we report the discovery of a
remarkable 0.3 sec X-ray pulsar, PSR J1846-0258, associated with the supernova
remnant Kes 75. With a characteristic age of only 723 yr, consistent with the
age of Kes 75, PSR J1846-0258 is the youngest pulsar yet discovered and is
being rapidly spun down by torques from a large magnetic dipole of strength ~
5E13 G, just above the so-called quantum critical field. PSR J1846-0258 resides
in this transitional regime where the magnetic field is hypothesized to
separate the regular pulsars from the so-called magnetars. PSR J1846-0258 is
evidently a Crab-like pulsar, however, its period, spin-down rate, spin-down
conversion efficiency, are each an order-of-magnitude greater, likely the
result of its extreme magnetic field.Comment: 4 pages, 3 figures, LaTex, emulateapj.sty. Submitted to The
Astrophysical Journa
Hopf algebras in dynamical systems theory
The theory of exact and of approximate solutions for non-autonomous linear
differential equations forms a wide field with strong ties to physics and
applied problems. This paper is meant as a stepping stone for an exploration of
this long-established theme, through the tinted glasses of a (Hopf and
Rota-Baxter) algebraic point of view. By reviewing, reformulating and
strengthening known results, we give evidence for the claim that the use of
Hopf algebra allows for a refined analysis of differential equations. We
revisit the renowned Campbell-Baker-Hausdorff-Dynkin formula by the modern
approach involving Lie idempotents. Approximate solutions to differential
equations involve, on the one hand, series of iterated integrals solving the
corresponding integral equations; on the other hand, exponential solutions.
Equating those solutions yields identities among products of iterated Riemann
integrals. Now, the Riemann integral satisfies the integration-by-parts rule
with the Leibniz rule for derivations as its partner; and skewderivations
generalize derivations. Thus we seek an algebraic theory of integration, with
the Rota-Baxter relation replacing the classical rule. The methods to deal with
noncommutativity are especially highlighted. We find new identities, allowing
for an extensive embedding of Dyson-Chen series of time- or path-ordered
products (of generalized integration operators); of the corresponding Magnus
expansion; and of their relations, into the unified algebraic setting of
Rota-Baxter maps and their inverse skewderivations. This picture clarifies the
approximate solutions to generalized integral equations corresponding to
non-autonomous linear (skew)differential equations.Comment: International Journal of Geometric Methods in Modern Physics, in
pres
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