14,839 research outputs found
Lyapunov Conditions for Differentiability of Markov Chain Expectations: the Absolutely Continuous Case
We consider a family of Markov chains whose transition dynamics are affected
by model parameters. Understanding the parametric dependence of (complex)
performance measures of such Markov chains is often of significant interest.
The derivatives of the performance measures w.r.t. the parameters play
important roles, for example, in numerical optimization of the performance
measures, and quantification of the uncertainties in the performance measures
when there are uncertainties in the parameters from the statistical estimation
procedures. In this paper, we establish conditions that guarantee the
differentiability of various types of intractable performance measures---such
as the stationary and random horizon discounted performance measures---of
general state space Markov chains and provide probabilistic representations for
the derivatives
The LPM effect in sequential bremsstrahlung 2: factorization
The splitting processes of bremsstrahlung and pair production in a medium are
coherent over large distances in the very high energy limit, which leads to a
suppression known as the Landau-Pomeranchuk-Migdal (LPM) effect. In this paper,
we continue analysis of the case when the coherence lengths of two consecutive
splitting processes overlap (which is important for understanding corrections
to standard treatments of the LPM effect in QCD), avoiding soft-gluon
approximations. In particular, this paper analyzes the subtle problem of how to
precisely separate overlapping double splitting (e.g.\ overlapping double
bremsstrahlung) from the case of consecutive, independent bremsstrahlung (which
is the case that would be implemented in a Monte Carlo simulation based solely
on single splitting rates). As an example of the method, we consider the rate
of real double gluon bremsstrahlung from an initial gluon with various
simplifying assumptions (thick media; approximation; large ; and
neglect for the moment of processes involving 4-gluon vertices) and explicitly
compute the correction due to overlapping formation
times.Comment: 59 pages, 37 figures. The major changes from v1: new section I.A.4
added to give kinetic theory analogy to better explain the importance of the
subtraction defining Delta[d(Gamma)/dx dy]; new appendix F added to
compare/contrast with issues raised by Blaizot, Dominguez, Iancu, and
Mehtar-Tani [22
The LPM effect in sequential bremsstrahlung: dimensional regularization
The splitting processes of bremsstrahlung and pair production in a medium are
coherent over large distances in the very high energy limit, which leads to a
suppression known as the Landau-Pomeranchuk-Migdal (LPM) effect. Of recent
interest is the case when the coherence lengths of two consecutive splitting
processes overlap (which is important for understanding corrections to standard
treatments of the LPM effect in QCD). In previous papers, we have developed
methods for computing such corrections without making soft-gluon
approximations. However, our methods require consistent treatment of canceling
ultraviolet (UV) divergences associated with coincident emission times, even
for processes with tree-level amplitudes. In this paper, we show how to use
dimensional regularization to properly handle the UV contributions. We also
present a simple diagnostic test that any consistent UV regularization method
for this problem needs to pass.Comment: 59 pages, 8 figures [main change from v1: addition of the new
appendix B summarizing more about use of the i*epsilon prescription in
earlier work
Selected highlights from the study of mesons
We provide a brief review of recent progress in the study of mesons using
QCD's Dyson-Schwinger equations. Along the way we touch on aspects of
confinement and dynamical chiral symmetry breaking but in the main focus upon:
exact results for pseudoscalar mesons, including aspects of the eta-eta'
problem; a realisation that the so-called vacuum condensates are actually an
intrinsic, localised property of hadrons; an essentially nonperturbative
procedure for constructing a symmetry-preserving Bethe-Salpeter kernel, which
has enabled a demonstration that dressed-quarks possess momentum-dependent
anomalous chromo- and electromagnetic moments that are large at infrared
momenta, and resolution of a longstanding problem in understanding the
mass-splitting between rho- and a1-mesons such that they are now readily seen
to be parity partners in the meson spectrum; features of electromagnetic form
factors connected with charged and neutral pions; and computation and
explanation of valence-quark distribution functions in pseudoscalar mesons. We
argue that in solving QCD, a constructive feedback between theory and extant
and forthcoming experiments will enable constraints to be placed on the
infrared behaviour of QCD's beta-function, the nonperturbative quantity at the
core of hadron physics.Comment: 28 pages, 15 figures, 2 tables. Version to appear in the Chinese
Journal of Physic
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