708 research outputs found
Faster generation of random spanning trees
In this paper, we set forth a new algorithm for generating approximately
uniformly random spanning trees in undirected graphs. We show how to sample
from a distribution that is within a multiplicative of uniform in
expected time \TO(m\sqrt{n}\log 1/\delta). This improves the sparse graph
case of the best previously known worst-case bound of , which has stood for twenty years.
To achieve this goal, we exploit the connection between random walks on
graphs and electrical networks, and we use this to introduce a new approach to
the problem that integrates discrete random walk-based techniques with
continuous linear algebraic methods. We believe that our use of electrical
networks and sparse linear system solvers in conjunction with random walks and
combinatorial partitioning techniques is a useful paradigm that will find
further applications in algorithmic graph theory
Energy spectra of gamma-rays, electrons and neutrinos produced at interactions of relativistic protons with low energy radiation
We derived simple analytical parametrizations for energy distributions of
photons, electrons, and neutrinos produced in interactions of relativistic
protons with an isotropic monochromatic radiation field. The results on
photomeson processes are obtained using numerical simulations of proton-photon
interactions based on the public available Monte-Carlo code SOPHIA. For
calculations of energy spectra of electrons and positrons from the pair
production (Bethe-Heitler) process we suggest a simple formalism based on the
well-known differential cross-section of the process in the rest frame of the
proton. The analytical presentations of energy distributions of photons and
leptons provide a simple but accurate approach for calculations of broad-band
energy spectra of gamma-rays and neutrinos in cosmic proton accelerators
located in radiation dominated environments.Comment: 17 pages, 21 figures, published in Phys.Rev.D. We have corrected two
misprints in the text. We note that the correct expressions were used for
calculations in the previous versions of the paper, thus the misprints did
not have an impact on the figure
Faster Approximate Multicommodity Flow Using Quadratically Coupled Flows
The maximum multicommodity flow problem is a natural generalization of the
maximum flow problem to route multiple distinct flows. Obtaining a
approximation to the multicommodity flow problem on graphs is a well-studied
problem. In this paper we present an adaptation of recent advances in
single-commodity flow algorithms to this problem. As the underlying linear
systems in the electrical problems of multicommodity flow problems are no
longer Laplacians, our approach is tailored to generate specialized systems
which can be preconditioned and solved efficiently using Laplacians. Given an
undirected graph with m edges and k commodities, we give algorithms that find
approximate solutions to the maximum concurrent flow problem and
the maximum weighted multicommodity flow problem in time
\tilde{O}(m^{4/3}\poly(k,\epsilon^{-1}))
On the spectral shape of radiation due to Inverse Compton Scattering close to the maximum cut-off
The spectral shape of radiation due to Inverse Compton Scattering is
analyzed, in the Thomson and the Klein-Nishina regime, for electron
distributions with exponential cut-off. We derive analytical, asymptotic
expressions for the spectrum close to the maximum cut-off region. We consider
monoenergetic, Planckian and Synchrotron photons as target photon fields. These
approximations provide a direct link between the distribution of parent
electrons and the up-scattered spectrum at the cut-off region.Comment: 27 pages, 15 figures, accepted for publication in Ap
Synchro-curvature radiation of charged particles in the strong curved magnetic fields
It is generally believed that the radiation of relativistic particles in a
curved magnetic field proceeds in either the synchrotron or the curvature
radiation modes. In this paper we show that in strong curved magnetic fields a
significant fraction of the energy of relativistic electrons can be radiated
away in the intermediate, the so-called synchro-curvature regime. Because of
the persistent change of the trajectory curvature, the radiation varies with
the frequency of particle gyration. While this effect can be ignored in the
synchrotron and curvature regimes, the variability plays a key role in the
formation of the synchro-curvature radiation. Using the Hamiltonian formalism,
we find that the particle trajectory has the form of a helix wound around the
drift trajectory. This allows us to calculate analytically the intensity and
energy distribution of prompt radiation in the general case of magnetic
bremsstrahlung in the curved magnetic field. We show that the transition to the
limit of the synchrotron and curvature radiation regimes is determined by the
relation between the drift velocity and the component of the particle velocity
perpendicular to the drift trajectory. The detailed numerical calculations,
which take into account the energy losses of particles, confirm the principal
conclusions based on the simplified analytical treatment of the problem, and
allow us to analyze quantitatively the transition between different radiation
regimes for a broad range of initial pitch angles. We argue that in the case of
realization of specific configurations of the electric and magnetic fields, the
gamma-ray emission of the pulsar magnetospheres can be dominated by the
component radiated in the synchro-curvature regime.Comment: this article supersedes arXiv:1207.6903 and arXiv:1305.078
On transition of propagation of relativistic particles from the ballistic to the diffusion regime
A stationary distribution function that describes the entire processes of
propagation of relativistic particles, including the transition between the
ballistic and diffusion regimes, is obtained. The spacial component of the
constructed function satisfies to the first two moments of the Boltzmann
equation. The angular part of the distribution provides accurate values for the
angular moments derived from the Boltzmann equation, and gives a correct
expression in the limit of small-angle approximation. Using the derived
function, we studied the gamma-ray images produced through the interaction
of relativistic particles with gas clouds in the proximity of the accelerator.
In general, the morphology and the energy spectra of gamma-rays significantly
deviate from the "standard" results corresponding to the propagation of
relativistic particles strictly in the diffusion regime
Mechanics and kinetics in the Friedmann-Lemaitre-Robertson-Walker space-times
Using the standard canonical formalism, the equations of mechanics and
kinetics in the Friedmann-Lemaitre-Robertson-Walker (FLRW) space-times in
Cartesian coordinates have been obtained. The transformation law of the
generalized momentum under the shift of the origin of the coordinate system has
been found, and the form invariance of the Hamiltonian function relative to the
shift transformation has been proved. The general solution of the collisionless
Boltzmann equation has been found. In the case of the homogeneous distribution
the solutions of the kinetic equation for several simple, but important for
applications, cases have been obtained
Simple analytical approximations for treatment of inverse Compton scattering of relativistic electrons in the black-body radiation field
The inverse Compton (IC) scattering of relativistic electrons is one of the
major gamma-ray production mechanisms in different environments. Often the
target photons for the IC scattering are dominated by black (or grey) body
radiation. In this case, the precise treatment of the characteristics of IC
radiation requires numerical integrations over the Planckian distribution.
Formally, analytical integrations are also possible but they result in series
of several special functions; this limits the efficiency of usage of these
expressions. The aim of this work is the derivation of approximate analytical
presentations which would provide adequate accuracy for the calculations of the
energy spectra of up-scattered radiation, the rate of electron energy losses,
and the mean energy of emitted photons. Such formulae have been obtained by
merging the analytical asymptotic limits. The coefficients in these expressions
are calculated via the least square fitting of the results of numerical
integrations. The simple analytical presentations, obtained for both the
isotropic and anisotropic target radiation fields, provide adequate (as good as
) accuracy for broad astrophysical applications.Comment: 16 pages, 11 figures, accepted for publication in Ap
Almost Optimal Streaming Algorithms for Coverage Problems
Maximum coverage and minimum set cover problems --collectively called
coverage problems-- have been studied extensively in streaming models. However,
previous research not only achieve sub-optimal approximation factors and space
complexities, but also study a restricted set arrival model which makes an
explicit or implicit assumption on oracle access to the sets, ignoring the
complexity of reading and storing the whole set at once. In this paper, we
address the above shortcomings, and present algorithms with improved
approximation factor and improved space complexity, and prove that our results
are almost tight. Moreover, unlike most of previous work, our results hold on a
more general edge arrival model. More specifically, we present (almost) optimal
approximation algorithms for maximum coverage and minimum set cover problems in
the streaming model with an (almost) optimal space complexity of
, i.e., the space is {\em independent of the size of the sets or
the size of the ground set of elements}. These results not only improve over
the best known algorithms for the set arrival model, but also are the first
such algorithms for the more powerful {\em edge arrival} model. In order to
achieve the above results, we introduce a new general sketching technique for
coverage functions: This sketching scheme can be applied to convert an
-approximation algorithm for a coverage problem to a
(1-\eps)\alpha-approximation algorithm for the same problem in streaming, or
RAM models. We show the significance of our sketching technique by ruling out
the possibility of solving coverage problems via accessing (as a black box) a
(1 \pm \eps)-approximate oracle (e.g., a sketch function) that estimates the
coverage function on any subfamily of the sets
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