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The art gallery theorem asserts that any polygon with n vertices can be protected by at most [n/3] stationary guards. The original proof by Chvátal uses a nonroutine and nonintuitive induction. We give a simple inductive proof of a new, more general result, the constrained art gallery theorem: If V∗and E∗are specified sets of vertices and edges that must contain guards, then the polygon can be protected by at most [(n + 2|V∗| + |E∗|) /3] guards. Our result reduces to Chvátal's art gallery theorem when V∗and E∗are empty. We give a second short proof of this generalization in the spirit of Fisk's proof of the art gallery theorem using graph colorings. © THE MATHEMATICAL ASSOCIATION OF AMERICA.

You will not see Robert Patterson's name mentioned in many mathematics books. While his mathematical works survive, his name is more likely to appear in American history books dealing with the Colonial period, given his associations with the most influential men of that time. In this article, we will examine his mathematical work, as well as his contributions to a newly-formed nation. Most of what we know about Robert Patterson's ancestors and life is due to his grandson, William Ewing DuBois, who wrote a family history in 1847. For other information, I have drawn upon diaries and a great many letters. All spelling and syntax are copied exactly as they appear. © 2015 British Society for the History of Mathematics.

To ensure the function of wireless sensor networks (WSNs), nodes that fail to forward packets must be localized efficiently and then fixed or replaced promptly. The state-of-the-art work frames lossy node localization in WSNs as an optimal sequential testing problem guided by end-to-end data. It combines both the active and passive measurements to minimize the testing cost and the number of iterations. However, this hybrid approach has many limitations. Inspired by the success of coverage-based software debugging, and the similarity between software debugging and lossy node localization, we propose a coverage-based lossy node detection for WSNs. Supported by established statistic theories, this approach greatly boosts the performance. Experiments on randomly generated networks and deployed networks show that the proposed algorithm can significantly reduce testing cost and number of iterations, which are the two optimization goals of previous work. We expect to use this approach for other diagnostic problems in WSNs. © 2001-2012 IEEE.

Triangulations of 3-dimensional polyhedron are partitions of the polyhedron with tetrahedra in a face-to-face fashion without introducing new vertices. Schönhardt (Math. Ann. 89:309–312, 1927), Bagemihl (Amer. Math. Mon. 55:411–413, 1948), Kuperberg (Personal communication 2011) and others constructed special polyhedra in such a way that clever one line geometric reasons imply nontriangulability. Rambau (Comb. Comput. Geom. 52:501–516, 2005) proved that twisted prisms over n-gons are nontriangulable. Our approach for proving polyhedra are nontriangulable is to show that partitions with tetrahedra, which we call tilings, do not exist even if the face-to-face-restriction is relaxed. First we construct a polyhedron which is tileable but is not triangulable. Then we revisit Rambau type twisted prisms. In fact we consider a slightly different class of polyhedra, and prove that these new twisted prisms are nontileable, thus are nontriangulable. We also show that one can twist the regular dodecahedron so that it becomes nontileable, which is abstracted to a new family of nontileable polyhedra, called nonconvex twisted pentaprisms. © 2015, The Managing Editors.

We study transformations of finite modules over Noetherian local rings that attach to a module M a graded module H(x)(M) defined via partial systems of parameters x of M. Despite the generality of the process, which are called j-transforms, in numerous cases they have interesting cohomological properties. We focus on deriving the Hilbert functions of j-transforms and studying the significance of the vanishing of some of its coefficients. Copyright © 2016 Cambridge Philosophical Society.

Motor vehicle accidents (MVA) are often difficult to distinguish from non-accidental injury (NAI). This retrospective case–control study compared animals with known MVA trauma against those with known NAI. Medical records of 426 dogs and cats treated after MVA and 50 after NAI were evaluated. Injuries significantly associated with MVA were pelvic fractures, pneumothorax, pulmonary contusion, abrasions, and degloving wounds. Injuries associated with NAI were fractures of the skull, teeth, vertebrae, and ribs, scleral hemorrhage, damage to claws, and evidence of older fractures. Odds ratios are reported for these injuries. MVA rib fractures were found to occur in clusters on one side of the body, with cranial ribs more likely to fracture, while NAI rib fractures were found to occur bilaterally with no cranial–caudal pattern. Establishing evidence-based patterns of injury may help clinicians differentiate causes of trauma and may aid in the documentation and prosecution of animal abuse. © 2016 American Academy of Forensic Sciences

We extend and unify most known results about guarding orthogonal polygons by introducing the same-sign diagonal graphs of a convex quadrangulation and applying results about vertex covers for graphs. Our approach also yields new theorems and often guarantees two disjoint vertex guard sets of relatively small cardinality. For instance, an orthogonal polygon on n vertices has two disjoint vertex guard sets of cardinality at most (Formula presented.). We give new proofs of Aggarwal’s one-hole theorem and the orthogonal fortress theorem. We prove that an orthogonal polygon with n vertices and any number of holes can be protected by at most (Formula presented.) vertex guards, improving the best known bound of (Formula presented.). Also, an orthogonal polygon with n vertices and h holes can be protected by (Formula presented.) guarded guards, which is best possible when (Formula presented.). Moreover, for orthogonal fortresses with n vertices, (Formula presented.) guarded guards are always sufficient and sometimes necessary. © 2016, Springer Science+Business Media New York (outside the USA).

Let P be an orthogonal polygon with n vertices, and let V⁎ and E⁎ be specified sets of vertices and edges of P. We prove that P has a guard set of cardinality at most ⌊(n+3|V⁎|+2|E⁎|)/4⌋ that includes each vertex in V⁎ and at least one point of each edge in E⁎. Our bound is sharp and reduces to the orthogonal art gallery theorem of Kahn, Klawe and Kleitman when V⁎ and E⁎ are empty. © 2016 Elsevier B.V.

Software components, which are vulnerable to being exploited, need to be identified and patched. Employing any prevention techniques designed for the purpose of detecting vulnerable software components in early stages can reduce the expenses associated with the software testing process significantly and thus help building a more reliable and robust software system. Although previous studies have demonstrated the effectiveness of adapting prediction techniques in vulnerability detection, the feasibility of those techniques is limited mainly because of insufficient training data sets. This paper proposes a prediction technique targeting at early identification of potentially vulnerable software components. In the proposed scheme, the potentially vulnerable components are viewed as mislabeled data that may contain true but not yet observed vulnerabilities. The proposed hybrid technique combines the supports vector machine algorithm and ensemble learning strategy to better identify potential vulnerable components. The proposed vulnerability detection scheme is evaluated using some Java Android applications. The results demonstrated that the proposed hybrid technique could identify potentially vulnerable classes with high precision and relatively acceptable accuracy and recall.