1,795 research outputs found
Nonlinear Instability for the Critically Dissipative Quasi-Geostrophic Equation
We prove that linear instability implies non-linear instability in the energy
norm for the critically dissipative quasi-geostrophic equation.Comment: 16 pages, corrected typos, a global bound that was obtained for the
unforced equation by Kiselev-Nazarov-Volberg obtained for the forced equation
and utilized in the paper
Nonequilibrium Phase Transitions in Models of Aggregation, Adsorption, and Dissociation
We study nonequilibrium phase transitions in a mass-aggregation model which
allows for diffusion, aggregation on contact, dissociation, adsorption and
desorption of unit masses. We analyse two limits explicitly. In the first case
mass is locally conserved whereas in the second case local conservation is
violated. In both cases the system undergoes a dynamical phase transition in
all dimensions. In the first case, the steady state mass distribution decays
exponentially for large mass in one phase, and develops an infinite aggregate
in addition to a power-law mass decay in the other phase. In the second case,
the transition is similar except that the infinite aggregate is missing.Comment: Major revision of tex
The Well-posedness of the Null-Timelike Boundary Problem for Quasilinear Waves
The null-timelike initial-boundary value problem for a hyperbolic system of
equations consists of the evolution of data given on an initial characteristic
surface and on a timelike worldtube to produce a solution in the exterior of
the worldtube. We establish the well-posedness of this problem for the
evolution of a quasilinear scalar wave by means of energy estimates. The
treatment is given in characteristic coordinates and thus provides a guide for
developing stable finite difference algorithms. A new technique underlying the
approach has potential application to other characteristic initial-boundary
value problems.Comment: Version to appear in Class. Quantum Gra
Support varieties for selfinjective algebras
Support varieties for any finite dimensional algebra over a field were
introduced by Snashall-Solberg using graded subalgebras of the Hochschild
cohomology. We mainly study these varieties for selfinjective algebras under
appropriate finite generation hypotheses. Then many of the standard results
from the theory of support varieties for finite groups generalize to this
situation. In particular, the complexity of the module equals the dimension of
its corresponding variety, all closed homogeneous varieties occur as the
variety of some module, the variety of an indecomposable module is connected,
periodic modules are lines and for symmetric algebras a generalization of
Webb's theorem is true
Einstein gravity as a 3D conformally invariant theory
We give an alternative description of the physical content of general
relativity that does not require a Lorentz invariant spacetime. Instead, we
find that gravity admits a dual description in terms of a theory where local
size is irrelevant. The dual theory is invariant under foliation preserving
3-diffeomorphisms and 3D conformal transformations that preserve the 3-volume
(for the spatially compact case). Locally, this symmetry is identical to that
of Horava-Lifshitz gravity in the high energy limit but our theory is
equivalent to Einstein gravity. Specifically, we find that the solutions of
general relativity, in a gauge where the spatial hypersurfaces have constant
mean extrinsic curvature, can be mapped to solutions of a particular gauge
fixing of the dual theory. Moreover, this duality is not accidental. We provide
a general geometric picture for our procedure that allows us to trade foliation
invariance for conformal invariance. The dual theory provides a new proposal
for the theory space of quantum gravity.Comment: 27 pages. Published version (minor changes and corrections
Huygens' Principle for the Klein-Gordon equation in the de Sitter spacetime
In this article we prove that the Klein-Gordon equation in the de Sitter
spacetime obeys the Huygens' principle only if the physical mass of the
scalar field and the dimension of the spatial variable are tied by
the equation . Moreover, we define the incomplete Huygens'
principle, which is the Huygens' principle restricted to the vanishing second
initial datum, and then reveal that the massless scalar field in the de Sitter
spacetime obeys the incomplete Huygens' principle and does not obey the
Huygens' principle, for the dimensions , only. Thus, in the de Sitter
spacetime the existence of two different scalar fields (in fact, with m=0 and
), which obey incomplete Huygens' principle, is equivalent to
the condition (in fact, the spatial dimension of the physical world). For
these two values of the mass are the endpoints of the so-called in
quantum field theory the Higuchi bound. The value of the
physical mass allows us also to obtain complete asymptotic expansion of the
solution for the large time. Keywords: Huygens' Principle; Klein-Gordon
Equation; de Sitter spacetime; Higuchi Boun
Lower Bounds for Heights in Relative Galois Extensions
The goal of this paper is to obtain lower bounds on the height of an
algebraic number in a relative setting, extending previous work of Amoroso and
Masser. Specifically, in our first theorem we obtain an effective bound for the
height of an algebraic number when the base field is a
number field and is Galois. Our second result
establishes an explicit height bound for any non-zero element which is
not a root of unity in a Galois extension , depending on
the degree of and the number of conjugates of
which are multiplicatively independent over . As a consequence, we
obtain a height bound for such that is independent of the
multiplicative independence condition
Generalized Smoluchowski equation with correlation between clusters
In this paper we compute new reaction rates of the Smoluchowski equation
which takes into account correlations. The new rate K = KMF + KC is the sum of
two terms. The first term is the known Smoluchowski rate with the mean-field
approximation. The second takes into account a correlation between clusters.
For this purpose we introduce the average path of a cluster. We relate the
length of this path to the reaction rate of the Smoluchowski equation. We solve
the implicit dependence between the average path and the density of clusters.
We show that this correlation length is the same for all clusters. Our result
depends strongly on the spatial dimension d. The mean-field term KMFi,j = (Di +
Dj)(rj + ri)d-2, which vanishes for d = 1 and is valid up to logarithmic
correction for d = 2, is the usual rate found with the Smoluchowski model
without correlation (where ri is the radius and Di is the diffusion constant of
the cluster). We compute a new rate: the correlation rate K_{i,j}^{C}
(D_i+D_j)(r_j+r_i)^{d-1}M{\big(\frac{d-1}{d_f}}\big) is valid for d \leq
1(where M(\alpha) = \sum+\infty i=1i\alphaNi is the moment of the density of
clusters and df is the fractal dimension of the cluster). The result is valid
for a large class of diffusion processes and mass radius relations. This
approach confirms some analytical solutions in d 1 found with other methods. We
also show Monte Carlo simulations which illustrate some exact new solvable
models
A smooth introduction to the wavefront set
The wavefront set provides a precise description of the singularities of a
distribution. Because of its ability to control the product of distributions,
the wavefront set was a key element of recent progress in renormalized quantum
field theory in curved spacetime, quantum gravity, the discussion of time
machines or quantum energy inequalitites. However, the wavefront set is a
somewhat subtle concept whose standard definition is not easy to grasp. This
paper is a step by step introduction to the wavefront set, with examples and
motivation. Many different definitions and new interpretations of the wavefront
set are presented. Some of them involve a Radon transform.Comment: 29 pages, 7 figure
Bounds for Bose-Einstein Correlation Functions
Bounds for the correlation functions of identical bosons are discussed for
the general case of a Gaussian density matrix. In particular, for a purely
chaotic system the two-particle correlation function must always be greater
than one. On the other hand, in the presence of a coherent component the
correlation function may take values below unity. The experimental situation is
briefly discussed.Comment: 7 pages, LaTeX, DMR-THEP-93-5/
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