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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 $(X,d,\mathrm{\mu <\#comment/>})$is a locally complete and separable metric measure space, then continuous functions are dense in the Newtonian space $N^{1,p}\left(X\right)$. Here the measure $\mathrm{\mu <\#comment/>}$is Borel and is finite and positive on all metric balls. In particular, we don’t assume properness of $X$, doubling of $\mathrm{\mu <\#comment/>}$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. 

ABSTRACT Spanning tree modulus is a generalization of effective resistance that is closely related to graph strength and fractional arboricity. The optimal edge density associated with the spanning tree modulus is known to produce two hierarchical decompositions of arbitrary graphs, one based on strength and the other on arboricity. Here we introduce an exact‐arithmetic algorithm for spanning tree modulus and the strength‐based decomposition using Cunningham's algorithm for graph vulnerability. The algorithm exploits an interesting connection between the spanning tree modulus and critical edge sets from the vulnerability problem. This paper introduces the new algorithm, describes a practical means for implementing it using integer arithmetic, and presents some examples and computational time scaling tests. 

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.