14,864 research outputs found
Nonlinear Stability of Relativistic Vortex Sheets in Three-Dimensional Minkowski Spacetime
We are concerned with the nonlinear stability of vortex sheets for the
relativistic Euler equations in three-dimensional Minkowski spacetime. This is
a nonlinear hyperbolic problem with a characteristic free boundary. In this
paper, we introduce a new symmetrization by choosing appropriate functions as
primary unknowns. A necessary and sufficient condition for the weakly linear
stability of relativistic vortex sheets is obtained by analyzing the roots of
the Lopatinski\u{\i} determinant associated to the constant coefficient
linearized problem. Under this stability condition, we show that the variable
coefficient linearized problem obeys an energy estimate with a loss of
derivatives. The construction of certain weight functions plays a crucial role
in absorbing error terms caused by microlocalization. Based on the weakly
linear stability result, we establish the existence and nonlinear stability of
relativistic vortex sheets under small initial perturbations by a Nash--Moser
iteration scheme.Comment: 105 pages; to appear in: Arch. Ration. Mech. Anal. 201
Weak Continuity of the Gauss-Codazzi-Ricci System for Isometric Embedding
We establish the weak continuity of the Gauss-Coddazi-Ricci system for
isometric embedding with respect to the uniform -bounded solution sequence
for , which implies that the weak limit of the isometric embeddings of the
manifold is still an isometric embedding. More generally, we establish a
compensated compactness framework for the Gauss-Codazzi-Ricci system in
differential geometry. That is, given any sequence of approximate solutions to
this system which is uniformly bounded in and has reasonable bounds on
the errors made in the approximation (the errors are confined in a compact
subset of ), then the approximating sequence has a weakly
convergent subsequence whose limit is a solution of the Gauss-Codazzi-Ricci
system. Furthermore, a minimizing problem is proposed as a selection criterion.
For these, no restriction on the Riemann curvature tensor is made
Dynamics of Order Parameter in Photoexcited Peierls Chain
The photoexcited dynamics of order parameter in Peierls chain is investigated
by using a microscopic quantum theory in the limit where the hot electrons may
establish themselves into a quasi-equilibrium state described by an effective
temperature. The optical phonon mode responsible for the Peierls instability is
coupled to the electron subsystem, and its dynamic equation is derived in terms
of the density matrix technique. Recovery dynamics of the order parameter is
obtained, which reveals a number of interesting features including the change
of oscillation frequency and amplitude at phase transition temperature and the
photo-induced switching of order parameter.Comment: 5 pages, 3 figure
i2MapReduce: Incremental MapReduce for Mining Evolving Big Data
As new data and updates are constantly arriving, the results of data mining
applications become stale and obsolete over time. Incremental processing is a
promising approach to refreshing mining results. It utilizes previously saved
states to avoid the expense of re-computation from scratch.
In this paper, we propose i2MapReduce, a novel incremental processing
extension to MapReduce, the most widely used framework for mining big data.
Compared with the state-of-the-art work on Incoop, i2MapReduce (i) performs
key-value pair level incremental processing rather than task level
re-computation, (ii) supports not only one-step computation but also more
sophisticated iterative computation, which is widely used in data mining
applications, and (iii) incorporates a set of novel techniques to reduce I/O
overhead for accessing preserved fine-grain computation states. We evaluate
i2MapReduce using a one-step algorithm and three iterative algorithms with
diverse computation characteristics. Experimental results on Amazon EC2 show
significant performance improvements of i2MapReduce compared to both plain and
iterative MapReduce performing re-computation
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