2,718 research outputs found

### Spherically Symmetric Gravitational Collapse of Perfect Fluids

Formulating a perfect fluid filled spherically symmetric metric utilizing the
3+1 formalism for general relativity, we show that the metric coefficients are
completely determined by the mass-energy distribution, and its time rate of
change on an initial spacelike hypersurface. Rather than specifying
Schwarzschild coordinates for the exterior of the collapsing region, we let the
interior dictate the form of the solution in the exterior, and thus both
regions are found to be written in one coordinate patch. This not only
alleviates the need for complicated matching schemes at the interface, but also
finds a new coordinate system for the Schwarzschild spacetime expressed in
generalized Painleve-Gullstrand coordinates.Comment: 3 pages, To appear in the proceedings of the eleventh Marcel
Grossmann meeting on general relativity (MGXI), 23-29 July, 2006, Berli

### Bose Gases Near Unitarity

We study the properties of strongly interacting Bose gases at the density and
temperature regime when the three-body recombination rate is substantially
reduced. In this regime, one can have a Bose gas with all particles in
scattering states (i.e. the "upper branch") with little loss even at unitarity
over the duration of the experiment. We show that because of bosonic
enhancement, pair formation is shifted to the atomic side of the original
resonance (where scattering length $a_s<0$), opposite to the fermionic case. In
a trap, a repulsive Bose gas remains mechanically stable when brought across
resonance to the atomic side until it reaches a critical scattering length
$a_{s}^{\ast}<0$. For $a_s<a_{s}^{\ast}$, the density consists of a core of
upper branch bosons surrounded by an outer layer of equilibrium phase. The
conditions of low three-body recombination requires that the particle number
$N<\alpha (T/\omega)^{5/2}$ in a harmonic trap with frequency $\omega$, where
$\alpha$ is a constant.Comment: 4 pages, 4 figure

### More "normal" than normal: scaling distributions and complex systems

One feature of many naturally occurring or engineered complex systems is tremendous variability in event sizes. To account for it, the behavior of these systems is often described using power law relationships or scaling distributions, which tend to be viewed as "exotic" because of their unusual properties (e.g., infinite moments). An alternate view is based on mathematical, statistical, and data-analytic arguments and suggests that scaling distributions should be viewed as "more normal than normal". In support of this latter view that has been advocated by Mandelbrot for the last 40 years, we review in this paper some relevant results from probability theory and illustrate a powerful statistical approach for deciding whether the variability associated with observed event sizes is consistent with an underlying Gaussian-type (finite variance) or scaling-type (infinite variance) distribution. We contrast this approach with traditional model fitting techniques and discuss its implications for future modeling of complex systems

### Cross-layer optimization in TCP/IP networks

TCP-AQM can be interpreted as distributed primal-dual algorithms to maximize aggregate utility over source rates. We show that an equilibrium of TCP/IP, if exists, maximizes aggregate utility over both source rates and routes, provided congestion prices are used as link costs. An equilibrium exists if and only if this utility maximization problem and its Lagrangian dual have no duality gap. In this case, TCP/IP incurs no penalty in not splitting traffic across multiple paths. Such an equilibrium, however, can be unstable. It can be stabilized by adding a static component to link cost, but at the expense of a reduced utility in equilibrium. If link capacities are optimally provisioned, however, pure static routing, which is necessarily stable, is sufficient to maximize utility. Moreover single-path routing again achieves the same utility as multipath routing at optimality

### Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)

Although the ``scale-free'' literature is large and growing, it gives neither
a precise definition of scale-free graphs nor rigorous proofs of many of their
claimed properties. In fact, it is easily shown that the existing theory has
many inherent contradictions and verifiably false claims. In this paper, we
propose a new, mathematically precise, and structural definition of the extent
to which a graph is scale-free, and prove a series of results that recover many
of the claimed properties while suggesting the potential for a rich and
interesting theory. With this definition, scale-free (or its opposite,
scale-rich) is closely related to other structural graph properties such as
various notions of self-similarity (or respectively, self-dissimilarity).
Scale-free graphs are also shown to be the likely outcome of random
construction processes, consistent with the heuristic definitions implicit in
existing random graph approaches. Our approach clarifies much of the confusion
surrounding the sensational qualitative claims in the scale-free literature,
and offers rigorous and quantitative alternatives.Comment: 44 pages, 16 figures. The primary version is to appear in Internet
Mathematics (2005

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