1,186 research outputs found
Fourier Transforms of Lorentz Invariant Functions
Fourier transforms of Lorentz invariant functions in Minkowski space, with
support on both the timelike and the spacelike domains are performed by means
of direct integration. The cases of 1+1 and 1+2 dimensions are worked out in
detail, and the results for 1+n dimensions are given.Comment: 15 pages, 1 figur
Hierarchy of the Selberg zeta functions
We introduce a Selberg type zeta function of two variables which interpolates
several higher Selberg zeta functions. The analytic continuation, the
functional equation and the determinant expression of this function via the
Laplacian on a Riemann surface are obtained.Comment: 14 page
Univalent Foundations and the UniMath Library
We give a concise presentation of the Univalent Foundations of mathematics outlining the main ideas, followed by a discussion of the UniMath library of formalized mathematics implementing the ideas of the Univalent Foundations (section 1), and the challenges one faces in attempting to design a large-scale library of formalized mathematics (section 2). This leads us to a general discussion about the links between architecture and mathematics where a meeting of minds is revealed between architects and mathematicians (section 3). On the way our odyssey from the foundations to the "horizon" of mathematics will lead us to meet the mathematicians David Hilbert and Nicolas Bourbaki as well as the architect Christopher Alexander
A Solution to the Graceful Exit Problem in Pre-Big Bang Cosmology
We examine the string cosmology equations with a dilaton potential in the
context of the Pre-Big Bang Scenario with the desired scale factor duality, and
give a generic algorithm for obtaining solutions with appropriate evolutionary
properties. This enables us to find pre-big bang type solutions with suitable
dilaton behaviour that are regular at , thereby solving the graceful exit
problem. However to avoid fine tuning of initial data, an `exotic' equation of
state is needed that relates the fluid properties to the dilaton field. We
discuss why such an equation of state should be required for reliable dilaton
behaviour at late times.Comment: 16 pages LaTeX, 5 figures. To appear in Physical Review
On graviton production in braneworld cosmology
We study braneworlds in a five dimensional bulk, where cosmological expansion
is mimicked by motion through AdS. We show that the five dimensional
graviton reduces to the four dimensional one in the late time approximation of
such braneworlds. Inserting a fixed regulator brane far from the physical
brane, we investigate quantum graviton production due to the motion of the
brane. We show that the massive Kaluza-Klein modes decouple completely from the
massless mode and they are not generated at all in the limit where the
regulator brane position goes to infinity. In the low energy limit, the
massless four dimensional graviton obeys the usual 4d equation and is therefore
also not generated in a radiation-dominated universe.Comment: 9 pages, minor changes, references correcte
Recursion relations and branching rules for simple Lie algebras
The branching rules between simple Lie algebras and its regular (maximal)
simple subalgebras are studied. Two types of recursion relations for anomalous
relative multiplicities are obtained. One of them is proved to be the
factorized version of the other. The factorization property is based on the
existence of the set of weights specific for each injection. The
structure of is easily deduced from the correspondence between the
root systems of algebra and subalgebra. The recursion relations thus obtained
give rise to simple and effective algorithm for branching rules. The details
are exposed by performing the explicit decomposition procedure for injection.Comment: 15p.,LaTe
Near-Horizon Geometry and the Entropy of a Minimally Coupled Scalar Field in the Schwarzschild Black Hole
In this article, we will discuss a Lorentzian sector calculation of the
entropy of a minimally coupled scalar field in the Schwarzschild black hole
background using the brick wall model of 't Hooft. In the original article, the
WKB approximation was used for the modes that are globally stationary. In a
previous article, we found that the WKB quantization rule together with a
proper counting of the states, leads to a new expression of the scalar field
entropy which is not proportional to the area of the horizon. The expression of
the entropy is logarithmically divergent in the brick wall cut-off parameter in
contrast to an inverse power divergence obtained earlier. In this article, we
will consider the entropy for a thin shell of matter field of a given thickness
surrounding the black hole horizon. The thickness is chosen to be large
compared with the Planck length and is of the order of the atomic scale. When
expressed in terms of a covariant cut-off parameter, the entropy of a thin
shell of matter field of a given thickness and surrounding the horizon in the
Schwarzschild black hole background is given by an expression proportional to
the area of the black hole horizon. This leading order divergent term in the
cut-off parameter remains to be logarithmically divergent. The logarithmic
divergence is expected from the nature of the solution in the near-horizon
region. We will find that these discussions are significant in the context of
the continuation to the Euclidean sector and the corresponding regularization
schemes used to evaluate the thermodynamical properties of matter fields in
curved spaces. These are related with the geometric aspects of curved spaces.
The above discussions are also important in presence of cosmological event
horizon.Comment: 15 pages, A few discussions are added in Section:III, Published in
J.Phys.Soc.Japan, A brief version of Section:II was separately published in
Nucl.Phys.B [Nucl. Phys. B 814, 212 (2009)
Tachyonic perturbations in AdS orbifolds
We show that scalar as well as vector and tensor metric perturbations in the
Randall-Sundrum II braneworld allow normalizable tachyonic modes, i.e.,
possible instabilities. These instabilities require nonvanishing initial
anisotropic stresses on the brane. We show with a specific example that within
the Randall-Sundrum II model, even though the tachyonic modes are excited, no
instability develops. We argue, however, that in the cosmological context
instabilities might in principle be present. We conjecture that the tachyonic
modes are due to the singularity of the orbifold construction. We illustrate
this with a simple but explicit toy model.Comment: 11 pages, matches published versio
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