We introduce a new troubled-cell indicator for the discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws. This indicator can be defined on unstructured meshes for high order DG methods and depends only on data from the target cell and its immediate neighbors. It is able to identify shocks without PDE sensitive parameters to tune. Extensive one- and two-dimensional simulations on the hyperbolic systems of Euler equations indicate the good performance of this new troubled-cell indicator coupled with a simple minmod-type TVD limiter for the Runge-Kutta DG (RKDG) methods.

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Forums have been one of the most important internet services since the 21st century. However, Forum users have to receive information in a passive way currently. The topics in forums are ordered by last update time (last reply time), thus the information that a user is interested in may be overwhelmed by a large number of other information. Users always have to scan many pages to find a minority of information they need. In this paper, based on the analysis of users' needs, I have designed a personalized forum topic ranking system. This personalized ranking system first calculates all the factors that will influence the user’s decision-making as to whether or not to view a topic by using his/her browsing history. Then the system predicts the click probability for each topic according to all the influence factors using a learned maximum entropy model. Finally, forum topics can be ranked by the predicted click probability, so as to make users find their favorite information easier. As shown by the experiments, the precision of the personalized ranking system is about 60% to 75%, which improves the traditional method (ordered by last update time) by 50% to 85%. In addition, with the normalization of the indicator functions of the maximum entropy model and the selection of the iterative endpoint, the training time can be lowered to an average of 0.01 seconds for each user. It indicates that such a model is able to meet the requirements of practical applications.

Denote by Mnthe set of n ×ncomplex matrices. Let f:Mn→[0, ∞)be a continuous map such that f(μUAU∗) =f(A)for any complex unit μ, A ∈Mnand unitary U∈Mn, f(X) =0if and only if X=0and the induced map t →f(tX)is monotonically increasing on [0, ∞)for any rank onenilpotent X∈Mn. Characterization is given for surjective maps φon Mnsatisfying f(AB−BA) =f(φ(A)φ(B) −φ(B)φ(A)). The general theorem isthen used to deduce results on special cases when the function is the pseudo spectrum and the pseudo spectral radius.

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