2,311 research outputs found
Moments in graphs
Let be a connected graph with vertex set and a {\em weight function}
that assigns a nonnegative number to each of its vertices. Then, the
{\em -moment} of at vertex is defined to be
M_G^{\rho}(u)=\sum_{v\in V} \rho(v)\dist (u,v) , where \dist(\cdot,\cdot)
stands for the distance function. Adding up all these numbers, we obtain the
{\em -moment of }: M_G^{\rho}=\sum_{u\in
V}M_G^{\rho}(u)=1/2\sum_{u,v\in V}\dist(u,v)[\rho(u)+\rho(v)]. This
parameter generalizes, or it is closely related to, some well-known graph
invariants, such as the {\em Wiener index} , when for every
, and the {\em degree distance} , obtained when
, the degree of vertex . In this paper we derive some
exact formulas for computing the -moment of a graph obtained by a general
operation called graft product, which can be seen as a generalization of the
hierarchical product, in terms of the corresponding -moments of its
factors. As a consequence, we provide a method for obtaining nonisomorphic
graphs with the same -moment for every (and hence with equal mean
distance, Wiener index, degree distance, etc.). In the case when the factors
are trees and/or cycles, techniques from linear algebra allow us to give
formulas for the degree distance of their product
Bubble fluctuations in inflation
In the context of the open inflationary universe, we calculate the amplitude
of quantum fluctuations which deform the bubble shape. These give rise to
scalar field fluctuations in the open Friedman-Robertson-Walker universe which
is contained inside the bubble. One can transform to a new gauge in which
matter looks perfectly smooth, and then the perturbations behave as tensor
modes (gravitational waves of very long wavelength). For , where
is the density parameter, the microwave temperature anisotropies
produced by these modes are of order . Here, is the expansion rate during inflation, is
the intrinsic radius of the bubble at the time of nucleation, is the
bubble wall tension and labels the different multipoles (). The
gravitational backreaction of the bubble has been ignored. In this
approximation, , and the new effect can be much larger than the
one due to ordinary gravitational waves generated during inflation (unless, of
course, gets too close to one, in which case the new effect
disappears).Comment: 17 pages, 3 figs, LaTeX, epsfig.sty, available at
ftp://ftp.ifae.es/preprint/ft/uabft387.p
Zero-th law in structural glasses: an example
We investigate the validity of a zeroth thermodynamic law for non-equilibrium
systems. In order to describe the thermodynamics of the glassy systems, it has
been introduced an extra parameter, the effective temperature which generalizes
the fluctuation-dissipation theorem (FDT) to off-equilibrium systems and
supposedly describes thermal fluctuations around the aging state. In particular
we analyze two coupled systems of harmonic oscillators with Monte Carlo
dynamics. We study in detail two types of dynamics: sequential dynamics, where
the coupling between the subsystems comes only from the Hamiltonian; and
parallel dynamics where there is another source of coupling: the dynamics. We
show how in the first case the effective temperatures of the two interacting
subsystems are different asymptotically due to the smallness of the thermal
conductivity in the aging regime. This explains why, in structural glasses,
different interacting degrees of freedom can stay at different effective
temperatures, and never thermalize.Comment: 10 pages. Contribution to the Proceedings of the ESF SPHINX meeting
`Glassy behaviour of kinetically constrained models' (Barcelona, March 22-25,
2001). To appear in a special issue of J. Phys. Cond. Mat
Gravity in the Randall-Sundrum Brane World
We discuss the weak gravitational field created by isolated matter sources in
the Randall-Sundrum brane-world. In the case of two branes of opposite tension,
linearized Brans-Dicke (BD) gravity is recovered on either wall, with different
BD parameters. On the wall with positive tension the BD parameter is larger
than 3000 provided that the separation between walls is larger than 4 times the
AdS radius. For the wall of negative tension, the BD parameter is always
negative but greater than -3/2. In either case, shadow matter from the other
wall gravitates upon us. For equal Newtonian mass, light deflection from shadow
matter is 25 % weaker than from ordinary matter. Hence, the effective mass of a
clustered object containing shadow dark matter would be underestimated if
naively measured through its lensing effect. For the case of a single wall of
positive tension, Einstein gravity is recovered on the wall to leading order,
and if the source is stationary the field stays localized near the wall. We
calculate the leading Kaluza-Klein corrections to the linearized gravitational
field of a non-relativistic spherical object and find that the metric is
different from the Schwarzschild solution at large distances. We believe that
our linearized solution corresponds to the field far from the horizon after
gravitational collapse of matter on the brane.Comment: 8 pages, 1 figure. Replaced with revised version to be published in
Phys. Rev. Lett. Some comments adde
Gravity Waves from Instantons
We perform a first principles computation of the spectrum of gravity waves
produced in open inflationary universes. The background spacetime is taken to
be the continuation of an instanton saddle point of the Euclidean no boundary
path integral. The two-point tensor correlator is computed directly from the
path integral and is shown to be unique and well behaved in the infrared. We
discuss the tensor contribution to the cosmic microwave background anisotropy
and show how it may provide an observational discriminant between different
types of primordial instantons.Comment: 19 pages, RevTex file, including two postscript figure file
Second Order Perturbations of a Macroscopic String; Covariant Approach
Using a world-sheet covariant formalism, we derive the equations of motion
for second order perturbations of a generic macroscopic string, thus
generalizing previous results for first order perturbations. We give the
explicit results for the first and second order perturbations of a contracting
near-circular string; these results are relevant for the understanding of the
possible outcome when a cosmic string contracts under its own tension, as
discussed in a series of papers by Vilenkin and Garriga. In particular, second
order perturbations are necessaary for a consistent computation of the energy.
We also quantize the perturbations and derive the mass-formula up to second
order in perturbations for an observer using world-sheet time . The high
frequency modes give the standard Minkowski result while, interestingly enough,
the Hamiltonian turns out to be non-diagonal in oscillators for low-frequency
modes. Using an alternative definition of the vacuum, it is possible to
diagonalize the Hamiltonian, and the standard string mass-spectrum appears for
all frequencies. We finally discuss how our results are also relevant for the
problems concerning string-spreading near a black hole horizon, as originally
discussed by Susskind.Comment: New discussion about the quantum mass-spectrum in chapter
- …