401 research outputs found
Sensitivity of wardrop equilibria
We study the sensitivity of equilibria in the well-known game theoretic traffic model due to Wardrop. We mostly consider single-commodity networks. Suppose, given a unit demand flow at Wardrop equilibrium, one increases the demand by Δ or removes an edge carrying only an Δ-fraction of flow. We study how the equilibrium responds to such an Δ-change.
Our first surprising finding is that, even for linear latency functions, for every Δ>â0, there are networks in which an Δ-change causes every agent to change its path in order to recover equilibrium. Nevertheless, we can prove that, for general latency functions, the flow increase or decrease on every edge is at most Δ.
Examining the latency at equilibrium, we concentrate on polynomial latency functions of degree at most p with nonnegative coefficients. We show that, even though the relative increase in the latency of an edge due to an Δ-change in the demand can be unbounded, the path latency at equilibrium increases at most by a factor of (1â+âΔ) p . The increase of the price of anarchy is shown to be upper bounded by the same factor. Both bounds are shown to be tight.
Let us remark that all our bounds are tight. For the multi-commodity case, we present examples showing that neither the change in edge flows nor the change in the path latency can be bounded
Weak solutions to problems involving inviscid fluids
We consider an abstract functional-differential equation derived from the
pressure-less Euler system with variable coefficients that includes several
systems of partial differential equations arising in the fluid mechanics. Using
the method of convex integration we show the existence of infinitely many weak
solutions for prescribed initial data and kinetic energy
SBV Regularity for Genuinely Nonlinear, Strictly Hyperbolic Systems of Conservation Laws in one space dimension
We prove that if is the entropy
solution to a strictly hyperbolic system of conservation laws with
genuinely nonlinear characteristic fields then up to a
countable set of times the function is in
, i.e. its distributional derivative is a measure with no
Cantorian part.
The proof is based on the decomposition of into waves belonging to
the characteristic families and the balance
of the continuous/jump part of the measures in regions bounded by
characteristics. To this aim, a new interaction measure \mu_{i,\jump} is
introduced, controlling the creation of atoms in the measure .
The main argument of the proof is that for all where the Cantorian part
of is not 0, either the Glimm functional has a downward jump, or there is
a cancellation of waves or the measure is positive
Stability and Instability of Extreme Reissner-Nordstr\"om Black Hole Spacetimes for Linear Scalar Perturbations I
We study the problem of stability and instability of extreme
Reissner-Nordstrom spacetimes for linear scalar perturbations. Specifically, we
consider solutions to the linear wave equation on a suitable globally
hyperbolic subset of such a spacetime, arising from regular initial data
prescribed on a Cauchy hypersurface crossing the future event horizon. We
obtain boundedness, decay and non-decay results. Our estimates hold up to and
including the horizon. The fundamental new aspect of this problem is the
degeneracy of the redshift on the event horizon. Several new analytical
features of degenerate horizons are also presented.Comment: 37 pages, 11 figures; published version of results contained in the
first part of arXiv:1006.0283, various new results adde
Hyperbolic Balance Laws with a Non Local Source
This paper is devoted to hyperbolic systems of balance laws with non local
source terms. The existence, uniqueness and Lipschitz dependence proved here
comprise previous results in the literature and can be applied to physical
models, such as Euler system for a radiating gas and Rosenau regularization of
the Chapman-Enskog expansion.Comment: 26 page
Exponential stability of the wave equation with memory and time delay
We study the asymptotic behaviour of the wave equation with viscoelastic
damping in presence of a time-delayed damping. We prove exponential stability
if the amplitude of the time delay term is small enough
On Charged Black Holes in Anti-de Sitter Space
We study the region inside the event horizon of charged black holes in five
dimensional asymptotically anti-de Sitter space, using as a probe two-sided
correlators which are dominated by spacelike geodesics penetrating the horizon.
The spacetimes we investigate include the Reissner-Nordstrom black hole and
perturbations thereof. The perturbed spacetimes can be found exactly, enabling
us to perform a local scan of the region between the inner and outer horizons.
Surprisingly, the two-sided correlators we calculate seem to be geometrically
protected from the instability of the inner horizon.Comment: 1+37 pages, 20 ps and eps figures, LaTeX. References added and
changes made to section
On the Nonlinear Stability of Asymptotically Anti-de Sitter Solutions
Despite the recent evidence that anti-de Sitter spacetime is nonlinearly
unstable, we argue that many asymptotically anti-de Sitter solutions are
nonlinearly stable. This includes geons, boson stars, and black holes. As part
of our argument, we calculate the frequencies of long-lived gravitational
quasinormal modes of AdS black holes in various dimensions. We also discuss a
new class of asymptotically anti-de Sitter solutions describing noncoalescing
black hole binaries.Comment: 26 pages. 5 figure
On the Use of Multipole Expansion in Time Evolution of Non-linear Dynamical Systems and Some Surprises Related to Superradiance
A new numerical method is introduced to study the problem of time evolution
of generic non-linear dynamical systems in four-dimensional spacetimes. It is
assumed that the time level surfaces are foliated by a one-parameter family of
codimension two compact surfaces with no boundary and which are conformal to a
Riemannian manifold C. The method is based on the use of a multipole expansion
determined uniquely by the induced metric structure on C. The approach is fully
spectral in the angular directions. The dynamics in the complementary 1+1
Lorentzian spacetime is followed by making use of a fourth order finite
differencing scheme with adaptive mesh refinement.
In checking the reliability of the introduced new method the evolution of a
massless scalar field on a fixed Kerr spacetime is investigated. In particular,
the angular distribution of the evolving field in to be superradiant scattering
is studied. The primary aim was to check the validity of some of the recent
arguments claiming that the Penrose process, or its field theoretical
correspondence---superradiance---does play crucial role in jet formation in
black hole spacetimes while matter accretes onto the central object. Our
findings appear to be on contrary to these claims as the angular dependence of
a to be superradiant scattering of a massless scalar field does not show any
preference of the axis of rotation. In addition, the process of superradiance,
in case of a massless scalar field, was also investigated. On contrary to the
general expectations no energy extraction from black hole was found even though
the incident wave packets was fine tuned to be maximally superradiant. Instead
of energy extraction the to be superradiant part of the incident wave packet
fails to reach the ergoregion rather it suffers a total reflection which
appears to be a new phenomenon.Comment: 49 pages, 11 figure
On the concentration of entropy for scalar conservation laws
We prove that the entropy for an -solution to a scalar conservation laws with continuous initial data is concentrated on a countably -rectifiable set. To prove this result we introduce the notion of Lagrangian representation of the solution and give regularity estimates on the solution
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