A Boundary Construction Algorithm for a Complex Planar Point Set

It is difficult to extract the boundary of complex planar points with nonuniform distribution of point density, concave envelopes, and holes. To solve this problem, an algorithm is proposed in this paper. Based on Delaunay triangulation, the maximum boundary angle threshold is introduced as the parameter in the extraction of the rough boundary. Then, the point looseness threshold is introduced, and the fine boundary extraction is conducted for the local areas such as concave envelopes and holes. Finally, the complete boundary result of the whole point set is obtained. The effectiveness of the proposed algorithm is verified by experiments on the simulated point set and practical measured point set. The experimental results indicate that it has wider applicability and more effectiveness in engineering applications than the state-of-the-art boundary construction algorithms based on Delaunay triangulation.

On the Jensen convex and Jensen concave envelopes of means

In recent papers, the convexity of quasiarithmetic means was characterized under twice differentiability assumptions. One of the main goals of this paper is to show that the convexity or concavity of a quasiarithmetic mean implies the twice continuous differentiability of its generator. As a consequence of this result, we can characterize those quasiarithmetic means which admit a lower convex and upper concave quasiarithmetic envelope.

Feedback-Capacity of Degraded Gaussian Vector BC using Directed Information and Concave Envelopes

It is known that the capacity region of a two user physically degraded discrete memoryless (DM) broadcastchannel (BC) is not enlarged by feedback. An identical result holds true for a physically degraded Gaussian BC,established later using a variant of the Entropy Power Inequality (EPI). In this paper, we extend the latter resultto a physically degraded Gaussian Vector BC (PD-GVBC). However, the extension is not EPI based, but employs arecent result on the factorization of concave envelopes. While the existing concave envelope factorization results donot hold in the presence of feedback, we show that factorizing the corresponding directed information quantitiessuffice to attain the feedback capacity region of a PD-GVBC. Our work demonstrates that factorizing concaveenvelopes of directed information can handle situations involving feedback. We further show that the capacityregion of a discrete memoryless reversely physically degraded BC is not enlarged by feedback.

Dynamics in the fundamental solution of a non-convex conservation law

There is a huge jump in the theory of conservation laws if the convexity assumption is dropped. We study a scalar conservation law without the convexity assumption by monitoring the dynamics in the fundamental solution. We introduce three shock types in addition to the usual genuine shock: left-, right- and double-sided contacts. There are three kinds of phenomenon for these shocks, called branching, merging and transforming. All of these shocks and phenomena can be observed if the flux function has two inflection points. A comprehensive picture of a global dynamics of a non-convex flux is discussed in terms of characteristic maps and dynamical convex–concave envelopes.