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A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently symmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in R^3, and his classification is equivalent to the converse for R^3. In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group.
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It is an amazing and a bit counter-intuitive discovery by Micha Perles from the sixties that there are ``non-rational polytopes: combinatorial types of convex polytopes that cannot be realized with rational vertex coordinates. We describe a simple construction of non-rational polytopes that does not need duality (Perles ``Gale diagrams): It starts from a non-rational point configuration in the plane, and proceeds with so-called Lawrence extensions. We also show that there are non-rational polyhedral surfaces in 3-space, a discovery by Ulrich Brehm from 1997. His construction also starts from any non-rational point configuration in the plane, and then performs what one should call Brehm extensions, in order to obtain non-rational partial surfaces. These examples and objects are first mile stones on the way to the remarkable universality theorems for polytopes and for polyhedral surfaces by Mnev (1986), Richter-Gebert (1994), and Brehm (1997).
Let $G$ be a 4-chromatic maximal planar graph (MPG) with the minimum degree of at least 4 and let $C$ be an even-length cycle of $G$.If $|f(C)|=2$ for every $f$ in some Kempe equivalence class of $G$, then we call $C$ an unchanged bichromatic cycle (UBC) of $G$, and correspondingly $G$ an unchanged bichromatic cycle maximal planar graph (UBCMPG) with respect to $C$, where $f(C)={f(v)| vin V(C)}$. For an UBCMPG $G$ with respect to an UBC $C$, the subgraph of $G$ induced by the set of edges belonging to $C$ and its interior (or exterior), denoted by $G^C$, is called a base-module of $G$; in particular, when the length of $C$ is equal to four, we use $C_4$ instead of $C$ and call $G^{C_4}$ a 4-base-module. In this paper, we first study the properties of UBCMPGs and show that every 4-base-module $G^{C_4}$ contains a 4-coloring under which $C_4$ is bichromatic and there are at least two bichromatic paths with different colors between one pair of diagonal vertices of $C_4$ (these paths are called module-paths). We further prove that every 4-base-module $G^{C_4}$ contains a 4-coloring (called decycle coloring) for which the ends of a module-path are colored by distinct colors. Finally, based on the technique of the contracting and extending operations of MPGs, we prove that 55-configurations and 56-configurations are reducible by converting the reducibility problem of these two classes of configurations into the decycle coloring problem of 4-base-modules.
Automated detection of software vulnerabilities is a fundamental problem in software security. Existing program analysis techniques either suffer from high false positives or false negatives. Recent progress in Deep Learning (DL) has resulted in a surge of interest in applying DL for automated vulnerability detection. Several recent studies have demonstrated promising results achieving an accuracy of up to 95% at detecting vulnerabilities. In this paper, we ask, how well do the state-of-the-art DL-based techniques perform in a real-world vulnerability prediction scenario?. To our surprise, we find that their performance drops by more than 50%. A systematic investigation of what causes such precipitous performance drop reveals that existing DL-based vulnerability prediction approaches suffer from challenges with the training data (e.g., data duplication, unrealistic distribution of vulnerable classes, etc.) and with the model choices (e.g., simple token-based models). As a result, these approaches often do not learn features related to the actual cause of the vulnerabilities. Instead, they learn unrelated artifacts from the dataset (e.g., specific variable/function names, etc.). Leveraging these empirical findings, we demonstrate how a more principled approach to data collection and model design, based on realistic settings of vulnerability prediction, can lead to better solutions. The resulting tools perform significantly better than the studied baseline: up to 33.57% boost in precision and 128.38% boost in recall compared to the best performing model in the literature. Overall, this paper elucidates existing DL-based vulnerability prediction systems potential issues and draws a roadmap for future DL-based vulnerability prediction research. In that spirit, we make available all the artifacts supporting our results: https://git.io/Jf6IA.
Combining the ideas of Riesz $s$-energy and $log$-energy, we introduce the so-called $s,log^t$-energy. In this paper, we investigate the asymptotic behaviors for $N,t$ fixed and $s$ varying of minimal $N$-point $s,log^t$-energy constants and configurations of an infinite compact metric space of diameter less than $1$. In particular, we study certain continuity and differentiability properties of minimal $N$-point $s,log^t$-energy constants in the variable $s$ and we show that in the limits as $srightarrow infty$ and as $srightarrow s_0>0,$ minimal $N$-point $s,log^t$-energy configurations tend to an $N$-point best-packing configuration and a minimal $N$-point $s_0,log^t$-energy configuration, respectively. Furthermore, the optimality of $N$ distinct equally spaced points on circles in $mathbb{R}^2$ for some certain $s,log^t$ energy problems was proved.
In recent work, G. E. Andrews and G. Simay prove a surprising relation involving parity palindromic compositions, and ask whether a combinatorial proof can be found. We extend their results to a more general class of compositions that are palindromic modulo $m$, that includes the parity palindromic case when $m=2$. We then provide combinatorial proofs for the cases $m=2$ and $m=3$.
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