13,572 research outputs found
Clifford Valued Differential Forms, and Some Issues in Gravitation, Electromagnetism and 'Unified' Theories
In this paper we show how to describe the general theory of a linear metric
compatible connection with the theory of Clifford valued differential forms.
This is done by realizing that for each spacetime point the Lie algebra of
Clifford bivectors is isomorphic to the Lie algebra of Sl(2,C). In that way the
pullback of the linear connection under a local trivialization of the bundle
(i.e., a choice of gauge) is represented by a Clifford valued 1-form. That
observation makes it possible to realize immediately that Einstein's
gravitational theory can be formulated in a way which is similar to a Sl(2,C)
gauge theory. Such a theory is compared with other interesting mathematical
formulations of Einstein's theory. and particularly with a supposedly "unified"
field theory of gravitation and electromagnetism proposed by M. Sachs. We show
that his identification of Maxwell equations within his formalism is not a
valid one. Also, taking profit of the mathematical methods introduced in the
paper we investigate a very polemical issue in Einstein gravitational theory,
namely the problem of the 'energy-momentum' conservation. We show that many
statements appearing in the literature are confusing or even wrong.Comment: Misprints and errors in some equations of the printed version have
been correcte
Large cities are less green
We study how urban quality evolves as a result of carbon dioxide emissions as
urban agglomerations grow. We employ a bottom-up approach combining two
unprecedented microscopic data on population and carbon dioxide emissions in
the continental US. We first aggregate settlements that are close to each other
into cities using the City Clustering Algorithm (CCA) defining cities beyond
the administrative boundaries. Then, we use data on emissions at a
fine geographic scale to determine the total emissions of each city. We find a
superlinear scaling behavior, expressed by a power-law, between
emissions and city population with average allometric exponent
across all cities in the US. This result suggests that the high productivity of
large cities is done at the expense of a proportionally larger amount of
emissions compared to small cities. Furthermore, our results are substantially
different from those obtained by the standard administrative definition of
cities, i.e. Metropolitan Statistical Area (MSA). Specifically, MSAs display
isometric scaling emissions and we argue that this discrepancy is due to the
overestimation of MSA areas. The results suggest that allometric studies based
on administrative boundaries to define cities may suffer from endogeneity bias
Fracturing the optimal paths
Optimal paths play a fundamental role in numerous physical applications
ranging from random polymers to brittle fracture, from the flow through porous
media to information propagation. Here for the first time we explore the path
that is activated once this optimal path fails and what happens when this new
path also fails and so on, until the system is completely disconnected. In fact
numerous applications can be found for this novel fracture problem. In the
limit of strong disorder, our results show that all the cracks are located on a
single self-similar connected line of fractal dimension .
For weak disorder, the number of cracks spreads all over the entire network
before global connectivity is lost. Strikingly, the disconnecting path
(backbone) is, however, completely independent on the disorder.Comment: 4 pages,4 figure
A worldwide model for boundaries of urban settlements
The shape of urban settlements plays a fundamental role in their sustainable
planning. Properly defining the boundaries of cities is challenging and remains
an open problem in the Science of Cities. Here, we propose a worldwide model to
define urban settlements beyond their administrative boundaries through a
bottom-up approach that takes into account geographical biases intrinsically
associated with most societies around the world, and reflected in their
different regional growing dynamics. The generality of the model allows to
study the scaling laws of cities at all geographical levels: countries,
continents, and the entire world. Our definition of cities is robust and holds
to one of the most famous results in Social Sciences: Zipf's law. According to
our results, the largest cities in the world are not in line with what was
recently reported by the United Nations. For example, we find that the largest
city in the world is an agglomeration of several small settlements close to
each other, connecting three large settlements: Alexandria, Cairo, and Luxor.
Our definition of cities opens the doors to the study of the economy of cities
in a systematic way independently of arbitrary definitions that employ
administrative boundaries
- …