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 $L^p$-bounded solution sequence for $p>2$, 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 $L^2$ and has reasonable bounds on the errors made in the approximation (the errors are confined in a compact subset of $H^{-1}_{\text{loc}}$), 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