Irrational guards are sometimes needed

In this paper we study the art gallery problem, which is one of the fundamental problems in computational geometry. The objective is to place a minimum number of guards inside a simple polygon such that the guards together can see the whole polygon. We say that a guard at position $x$ sees a point $y$ if the line segment $xy$ is fully contained in the polygon. Despite an extensive study of the art gallery problem, it remained an open question whether there are polygons given by integer coordinates that require guard positions with irrational coordinates in any optimal solution. We give a positive answer to this question by constructing a monotone polygon with integer coordinates that can be guarded by three guards only when we allow to place the guards at points with irrational coordinates. Otherwise, four guards are needed. By extending this example, we show that for every $n$, there is polygon which can be guarded by $3n$ guards with irrational coordinates but need $4n$ guards if the coordinates have to be rational. Subsequently, we show that there are rectilinear polygons given by integer coordinates that require guards with irrational coordinates in any optimal solution.Comment: 18 pages 10 Figure

An Upper Bound Theorem for a Class of Flag Weak Pseudomanifolds

If K is an odd-dimensional flag closed manifold, flag generalized homology sphere or a more general flag weak pseudomanifold with sufficiently many vertices, then the maximal number of edges in K is achieved by the balanced join of cycles. The proof relies on stability results from extremal graph theory. In the case of manifolds we also offer an alternative (very) short proof utilizing the non-embeddability theorem of Flores. The main theorem can also be interpreted without the topological contents as a graph-theoretic extremal result about a class of graphs such that 1) every maximal clique in the graph has size d+1 and 2) every clique of size d belongs to exactly two maximal cliques

Complexity and Inapproximability Results for Parallel Task Scheduling and Strip Packing

We study the Parallel Task Scheduling problem $Pm|size_j|C_{\max}$ with a constant number of machines. This problem is known to be strongly NP-complete for each $m \geq 5$, while it is solvable in pseudo-polynomial time for each $m \leq 3$. We give a positive answer to the long-standing open question whether this problem is strongly $NP$-complete for $m=4$. As a second result, we improve the lower bound of $\frac{12}{11}$ for approximating pseudo-polynomial Strip Packing to $\frac{5}{4}$. Since the best known approximation algorithm for this problem has a ratio of $\frac{4}{3} + \varepsilon$, this result narrows the gap between approximation ratio and inapproximability result by a significant step. Both results are proven by a reduction from the strongly $NP$-complete problem 3-Partition

Consensus Paper: Cerebellum and Emotion

Over the past three decades, insights into the role of the cerebellum in emotional processing have substantially increased. Indeed, methodological refinements in cerebellar lesion studies and major technological advancements in the field of neuroscience are in particular responsible to an exponential growth of knowledge on the topic. It is timely to review the available data and to critically evaluate the current status of the role of the cerebellum in emotion and related domains. The main aim of this article is to present an overview of current facts and ongoing debates relating to clinical, neuroimaging, and neurophysiological findings on the role of the cerebellum in key aspects of emotion. Experts in the field of cerebellar research discuss the range of cerebellar contributions to emotion in nine topics. Topics include the role of the cerebellum in perception and recognition, forwarding and encoding of emotional information, and the experience and regulation of emotional states in relation to motor, cognitive, and social behaviors. In addition, perspectives including cerebellar involvement in emotional learning, pain, emotional aspects of speech, and neuropsychiatric aspects of the cerebellum in mood disorders are briefly discussed. Results of this consensus paper illustrate how theory and empirical research have converged to produce a composite picture of brain topography, physiology, and function that establishes the role of the cerebellum in many aspects of emotional processing

a Função de Orientador Argumentativo dos Rotulos em Artigo de Opinião: uma Questão de Entonação de Valores Sociais

À luz de uma perspectiva sociointeracionista, este trabalho tem como objetivo pesquisar o processo referenciação através dos rótulos (expressões nominais que sumarizam porções de texto), evidenciando a função de orientador argumentativo desempenhada por essas expressões. Assumida a perspectiva teórica, entendemos que o real é fabricado através de nossas práticas sociais e discursivas. Assim, ao contrário de concebermos que a relação entre linguagem e mundo se dá de forma direta e especular, defendemos as versões públicas do mundo (MONDADA e DUBOIS 2003[1995]). A fim de buscar o objetivo proposto, trazemos para o nosso texto a concepção filosófica bakthiniana de valoração (BAKTHIN, 1993[1920-24]), com o intuito de mostrar que, ao construir o objeto de discurso, o sujeito seleciona as formas linguísticas que melhor se adaptem ao seu projeto de dizer. No entanto, essas escolhas não emanam de um sujeito solitário; pelo contrário, são usadas por um sujeito que é singular e social, simultaneamente, que rejeita e assume diversos valores sociais, os quais determinam a forma e o conteúdo de seu enunciado e com os quais convive. Os textos utilizados exemplificar o que é discutido nessa pesquisa são três artigos de opinião publicados no jornal A Gazeta, selecionados no período de agosto a dezembro de 2009. Para a análise dos textos tomamos uma postura dialógica a fim de explicitar que o posicionamento assumido pelo articulista na construção do objeto de discurso é dialógico, pois responde a outros enunciados e entoa uma avaliação que não emana de um sujeito isolado, mas social. Desse modo, acreditamos que a função de orientador argumentativo dos rótulos é mais bem compreendida se levarmos em consideração a corrente dialógica na qual se insere o uso dessas expressões. Palavras-chave: Referenciação; Objeto de discurso; Rotulação; Dialogia; Valoração

Reordering buffer management with advice

In the reordering buffer management problem, a sequence of colored items arrives at a service station to be processed. Each color change between two consecutively processed items generates some cost. A reordering buffer of capacity k items can be used to preprocess the input sequence in order to decrease the number of color changes. The goal is to find a scheduling strategy that, using the reordering buffer, minimizes the number of color changes in the given sequence of items. We consider the problem in the setting of online computation with advice. In this model, the color of an item becomes known only at the time when the item enters the reordering buffer. Additionally, together with each item entering the buffer, we get a fixed number of advice bits, which can be seen as information about the future or as information about an optimal solution (or an approximation thereof) for the whole input sequence. We show that for any ε>0 there is a (1+ε)-competitive algorithm for the problem which uses only a constant (depending on ε) number of advice bits per input item. This also immediately implies a (1+ε)-approximation algorithm which has 2O(nlog1/ε) running time (this should be compared to the trivial optimal algorithm which has a running time of kO(n)). We complement the above result by presenting a lower bound of Ω(logk) bits of advice per request for any 1-competitive algorithm

Supersymmetric lattice fermions on the triangular lattice: superfrustration and criticality

We study a model for itinerant, strongly interacting fermions where a judicious tuning of the interactions leads to a supersymmetric Hamiltonian. On the triangular lattice this model is known to exhibit a property called superfrustration, which is characterized by an extensive ground state entropy. Using a combination of numerical and analytical methods we study various ladder geometries obtained by imposing doubly periodic boundary conditions on the triangular lattice. We compare our results to various bounds on the ground state degeneracy obtained in the literature. For all systems we find that the number of ground states grows exponentially with system size. For two of the models that we study we obtain the exact number of ground states by solving the cohomology problem. For one of these, we find that via a sequence of mappings the entire spectrum can be understood. It exhibits a gapped phase at 1/4 filling and a gapless phase at 1/6 filling and phase separation at intermediate fillings. The gapless phase separates into an exponential number of sectors, where the continuum limit of each sector is described by a superconformal field theory.Comment: 50 pages, 12 figures, 2 appendice

Competitive Tendering In The Netherlands: Central Planning Or Functional Specifications?

Institute of Transport and Logistics Studies. Faculty of Economics and Business. The University of Sydne