748 research outputs found
Harmonic Analysis of Linear Fields on the Nilgeometric Cosmological Model
To analyze linear field equations on a locally homogeneous spacetime by means
of separation of variables, it is necessary to set up appropriate harmonics
according to its symmetry group. In this paper, the harmonics are presented for
a spatially compactified Bianchi II cosmological model -- the nilgeometric
model. Based on the group structure of the Bianchi II group (also known as the
Heisenberg group) and the compactified spatial topology, the irreducible
differential regular representations and the multiplicity of each irreducible
representation, as well as the explicit form of the harmonics are all
completely determined. They are also extended to vector harmonics. It is
demonstrated that the Klein-Gordon and Maxwell equations actually reduce to
systems of ODEs, with an asymptotic solution for a special case.Comment: 28 pages, no figures, revised version to appear in JM
A new proof of the Bianchi type IX attractor theorem
We consider the dynamics towards the initial singularity of Bianchi type IX
vacuum and orthogonal perfect fluid models with a linear equation of state. The
`Bianchi type IX attractor theorem' states that the past asymptotic behavior of
generic type IX solutions is governed by Bianchi type I and II vacuum states
(Mixmaster attractor). We give a comparatively short and self-contained new
proof of this theorem. The proof we give is interesting in itself, but more
importantly it illustrates and emphasizes that type IX is special, and to some
extent misleading when one considers the broader context of generic models
without symmetries.Comment: 26 pages, 5 figure
Monotonic functions in Bianchi models: Why they exist and how to find them
All rigorous and detailed dynamical results in Bianchi cosmology rest upon
the existence of a hierarchical structure of conserved quantities and monotonic
functions. In this paper we uncover the underlying general mechanism and derive
this hierarchical structure from the scale-automorphism group for an
illustrative example, vacuum and diagonal class A perfect fluid models. First,
kinematically, the scale-automorphism group leads to a reduced dynamical system
that consists of a hierarchy of scale-automorphism invariant sets. Second, we
show that, dynamically, the scale-automorphism group results in
scale-automorphism invariant monotone functions and conserved quantities that
restrict the flow of the reduced dynamical system.Comment: 26 pages, replaced to match published versio
On Languages Accepted by P/T Systems Composed of joins
Recently, some studies linked the computational power of abstract computing
systems based on multiset rewriting to models of Petri nets and the computation
power of these nets to their topology. In turn, the computational power of
these abstract computing devices can be understood by just looking at their
topology, that is, information flow.
Here we continue this line of research introducing J languages and proving
that they can be accepted by place/transition systems whose underlying net is
composed only of joins. Moreover, we investigate how J languages relate to
other families of formal languages. In particular, we show that every J
language can be accepted by a log n space-bounded non-deterministic Turing
machine with a one-way read-only input. We also show that every J language has
a semilinear Parikh map and that J languages and context-free languages (CFLs)
are incomparable
Restricted infinitesimal deformations of restricted simple Lie algebras
We compute the restricted infinitesimal deformations of the restricted simple
Lie algebras over an algebraically closed field of characteristic different
from 2 and 3.Comment: 15 pages; final version, to appear in Journal of Algebra and Its
Application
On dual canonical bases
The dual basis of the canonical basis of the modified quantized enveloping
algebra is studied, in particular for type . The construction of a basis for
the coordinate algebra of the quantum matrices is appropriate for
the study the multiplicative property. It is shown that this basis is invariant
under multiplication by certain quantum minors including the quantum
determinant. Then a basis of quantum SL(n) is obtained by setting the quantum
determinant to one. This basis turns out to be equivalent to the dual canonical
basis
Electroweak Corrections using Effective Field Theory: Applications to the LHC
Electroweak Sudakov logarithms at high energy, of the form alpha/sin^2
theta_W^n log^m s/M_{Z,W}^2, are summed using effective theory (EFT) methods.
The exponentiation of Sudakov logarithms and factorization is discussed in the
EFT formalism. Radiative corrections are computed to scattering processes in
the standard model involving an arbitrary number of external particles. The
computations include non-zero particle masses such as the t-quark mass,
electroweak mixing effects which lead to unequal W and Z masses and a massless
photon, and Higgs corrections proportional to the top quark Yukawa coupling.
The structure of the radiative corrections, and which terms are summed by the
EFT renormalization group is discussed in detail. The omitted terms are smaller
than 1%. We give numerical results for the corrections to dijet production,
dilepton production, t-\bar t production, and squark pair production. The
purely electroweak corrections are significant -- about 15% at 1 TeV,
increasing to 30% at 5 TeV, and they change both the scattering rate and
angular distribution. The QCD corrections (which are well-known) are also
computed with the EFT. They are much larger -- about a factor of four at 1 TeV,
increasing to a factor of thirty at 5 TeV. Mass effects are also significant;
the q \bar q -> t \bar t rate is enchanced relative to the light-quark
production rate by 40%.Comment: Additional details added on exponentiation, and the form of the
Sudakov series. Figures darkened to print better. 40 pages, 40 figure
Electrocardiogram of the Mixmaster Universe
The Mixmaster dynamics is revisited in a new light as revealing a series of
transitions in the complex scale invariant scalar invariant of the Weyl
curvature tensor best represented by the speciality index , which
gives a 4-dimensional measure of the evolution of the spacetime independent of
all the 3-dimensional gauge-dependent variables except for the time used to
parametrize it. Its graph versus time characterized by correlated isolated
pulses in its real and imaginary parts corresponding to curvature wall
collisions serves as a sort of electrocardiogram of the Mixmaster universe,
with each such pulse pair arising from a single circuit or ``complex pulse''
around the origin in the complex plane. These pulses in the speciality index
and their limiting points on the real axis seem to invariantly characterize
some of the so called spike solutions in inhomogeneous cosmology and should
play an important role as a gauge invariant lens through which to view current
investigations of inhomogeneous Mixmaster dynamics.Comment: version 3: 20 pages iopart style, 19 eps figure files for 8 latex
figures; added example of a transient true spike to contrast with the
permanent true spike example from the Lim family of true spike solutions;
remarks in introduction and conclusion adjusted and toned down; minor
adjustments to the remaining tex
Electroweak Sudakov Logarithms and Real Gauge-Boson Radiation in the TeV Region
Electroweak radiative corrections give rise to large negative,
double-logarithmically enhanced corrections in the TeV region. These are partly
compensated by real radiation and, moreover, affected by selecting
isospin-noninvariant external states. We investigate the impact of real gauge
boson radiation more quantitatively by considering different restricted final
state configurations. We consider successively a massive abelian gauge theory,
a spontaneously broken SU(2) theory and the electroweak Standard Model. We find
that details of the choice of the phase space cuts, in particular whether a
fraction of collinear and soft radiation is included, have a strong impact on
the relative amount of real and virtual corrections.Comment: 20 pages, 4 figure
General Relativistic 1+3 Orthonormal Frame Approach Revisited
The equations of the 1+3 orthonormal frame approach are explicitly presented
and discussed. Natural choices of local coordinates are mentioned. A
dimensionless formulation is subsequently given. It is demonstrated how one can
obtain a number of interesting problems by specializing the general equations.
In particular, equation systems for ``silent'' dust cosmological models also
containing magnetic Maxwell fields, locally rotationally symmetric spacetime
geometries and spatially homogeneous cosmological models are presented. We show
that while the 3-Cotton--York tensor is zero for Szekeres dust models, it is
nonzero for a generic representative within the ``silent'' class.Comment: 41 pages, uufiles encoded postscript file, submitted to Phys. Rev.
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