In our earlier work, we constructed uniformly high order accurate discontinuous Galerkin (DG) and finite volume schemes which preserve positivity of density and pressure for the Euler equations of compressible gas dynamics. In this paper, we present an extension of this framework to construct positivity-preserving high order essentially non-oscillatory (ENO) and weighted essentially non-oscillatory (WENO) finite difference schemes for compressible Euler equations. General equations of state and source terms are also discussed. Numerical tests of the fifth order finite difference WENO scheme are reported to demonstrate the good behavior of such schemes.

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The tip multifractal spectrum of a 2-dimensional curve is one way to describe the behavior of the uniformizing conformal map of the complement near the tip. We give the tip multifractal spectrum for a Schramm–Loewner evolution (SLE) curve, we prove that the spectrum is valid with probability 1, and we give applications to the scaling of harmonic measure at the tip.

3-D CAD models are an important digital resource in the manufacturing industry. 3-D CAD model retrieval has become a key technology in product lifecycle management enabling the reuse of existing design data. In this paper, we propose a new method to retrieve 3-D CAD models based on 2-D pen-based sketch inputs. Sketching is a common and convenient method for communicating design intent during early stages of product design, e.g., conceptual design. However, converting sketched information into precise 3-D engineering models is cumbersome, and much of this effort can be avoided by reuse of existing data. To achieve this purpose, we present a user-adaptive sketch-based retrieval method in this paper. The contributions of this work are twofold. First, we propose a statistical measure for CAD model retrieval: the measure is based on sketch similarity and accounts for users’ drawing habits. Second, for 3-D CAD models in the database, we propose a sketch generation pipeline that represents each 3-D CAD model by a small yet sufficient set of sketches that are perceptually similar to human drawings. User studies and experiments that demonstrate the effectiveness of the proposed method in the design process are presented.

We study isometric Lie group actions on symmetric spaces admitting a section, i.e., a submanifold that meets all orbits orthogonally at every intersection point. We classify such actions on the compact symmetric spaces with simple isometry group and rank greater than one. In particular, we show that these actions are hyperpolar, i.e., the sections are flat.