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We show that capacity can be computed with locally Lipschitz functions in locally complete and separable metric spaces. Further, we show that if$(X,d,\mu )$is a locally complete and separable metric measure space, then continuous functions are dense in the Newtonian space$N^{1,p}(X)$. Here the measure$\mu$is Borel and is finite and positive on all metric balls. In particular, we don’t assume properness of$X$, doubling of$\mu$or any Poincaré inequalities. These resolve, partially or fully, questions posed by a number of authors, including J. Heinonen, A. Björn and J. Björn. In contrast to much of the past work, our results apply tolocally completespaces$X$and dispenses with the frequently used regularity assumptions: doubling, properness, Poincaré inequality, Loewner property or quasiconvexity.

 

The proliferation of Android applications has resulted in many malicious apps entering the market and causing significant damage. Robust techniques that determine if an app is malicious are greatly needed. We propose the use of network-based approaches to effectively separate malicious from benign apps, based on a small labeled dataset. The apps in our dataset come from the Google Play Store and have been scanned for malicious behavior using VirusTotal to produce a ground truth dataset with labels malicious or benign. The apps in the resulting dataset have been represented in the form of binary feature vectors (where the features represent permissions, intent actions, discriminative APIs, obfuscation signatures, and native code signatures). We have used these vectors to build a weighted network that captures the “closeness” between apps. We propagate labels from the labeled apps to unlabeled apps, and evaluate the effectiveness of the approaches studied using the Fl-measure. We have conducted experiments to compare three variants of the label propagation approaches on datasets that consist of increasingly larger amounts of labeled data. 

The theory ofp‐modulus provides a general framework for quantifying the richness of a family of objects on a graph. When applied to the family of spanning trees,p‐modulus has an interesting probabilistic interpretation. In particular, the 2‐modulus problem in this case has been shown to be equivalent to the problem of finding a probability distribution on spanning trees that utilizes the edges of the graph as fairly as possible. In the present work, we use this fact to produce a game‐theoretic interpretation of modulus by employing modulus to solve a secure broadcast game.