1,482 research outputs found

    Dynamic Connectivity in Disk Graphs

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    Let S ⊆ R2 be a set of n sites in the plane, so that every site s ∈ S has an associated radius rs > 0. Let D(S) be the disk intersection graph defined by S, i.e., the graph with vertex set S and an edge between two distinct sites s, t ∈ S if and only if the disks with centers s, t and radii rs , rt intersect. Our goal is to design data structures that maintain the connectivity structure of D(S) as sites are inserted and/or deleted in S. First, we consider unit disk graphs, i.e., we fix rs = 1, for all sites s ∈ S. For this case, we describe a data structure that has O(log2 n) amortized update time and O(log n/ log log n) query time. Second, we look at disk graphs with bounded radius ratio Ψ, i.e., for all s ∈ S, we have 1 ≤ rs ≤ Ψ, for a parameter Ψ that is known in advance. Here, we not only investigate the fully dynamic case, but also the incremental and the decremental scenario, where only insertions or only deletions of sites are allowed. In the fully dynamic case, we achieve amortized expected update time O(Ψ log4 n) and query time O(log n/ log log n). This improves the currently best update time by a factor of Ψ. In the incremental case, we achieve logarithmic dependency on Ψ, with a data structure that has O(α(n)) amortized query time and O(log Ψ log4 n) amortized expected update time, where α(n) denotes the inverse Ackermann function. For the decremental setting, we first develop an efficient decremental disk revealing data structure: given two sets R and B of disks in the plane, we can delete disks from B, and upon each deletion, we receive a list of all disks in R that no longer intersect the union of B. Using this data structure, we get decremental data structures with a query time of O(log n/ log log n) that supports deletions in O(n log Ψ log4 n) overall expected time for disk graphs with bounded radius ratio Ψ and O(n log5 n) overall expected time for disk graphs with arbitrary radii, assuming that the deletion sequence is oblivious of the internal random choices of the data structures

    Glimmers from the Axiverse

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    We study axion-photon couplings in compactifications of type IIB string theory. We find that these couplings are systematically suppressed compared to the inverse axion periodicity, as a result of two effects. First, couplings to the QED theta angle are suppressed for axion mass eigenstates that are light compared to the mass scale set by stringy instantons on the cycle supporting QED. Second, in compactifications with many axions the intersection matrix is sparse, making kinetic mixing weak. We study the resulting phenomenology in an ensemble of 200,000200{,}000 toy models constructed from the Kreuzer-Skarke database up to the maximum Hodge number h1,1=491h^{1,1}=491. We examine freeze-in production and decay of thermal axions, birefringence of the cosmic microwave background, X-ray spectrum oscillations, and constraints on the QCD axion from supernovae. We conclude that compactifications in this corner of the landscape involve many invisible axions, as well as a handful that may be detectable via photon couplings.Comment: 46 pages, 18 Figures, one appendi

    Topographic control of order-disorder phase transitions in a quasi-2D granular system

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    The focus of current research in two-dimensional phase transitions has shifted towards non-equilibrium systems such as active matter and fluid dynamics. However, unlike in equilibrium systems, we lack a complete framework to describe their behaviour. Although previous work has shown that some basic concepts from statistical mechanics can be applied to non-equilibrium systems, the extent to which they can be applied remains unclear. One intriguing problem in equilibrium systems is the two-dimensional hard-disc liquid-to-crystal phase transition. The nature of this phase transition differs from that in three-dimensions and was, until recently, a matter of much debate. Extending this debate, two-dimensional granular systems have also been studied to investigate the applicability of hard-disc model descriptions to non-equilibrium systems. Granular systems are convenient for manipulation and offer easy observations at the particle level and therefore represent an ideal test case for these investigations. In this thesis, I present an investigation of the order-disorder phase transition in a 2D driven granular system. Previous research has shown that these systems undergo a continuous two-step phase transition. We explore a mechanism for changing the nature of this transition from continuous to first-order by introducing a triangular lattice of dimples milled into the surface. The change in phase transition behaviour, for the system we focus on for much of this thesis, enables further study of other behaviours from equilibrium physics, such as hysteresis, surface tension and wetting. The phase behaviour of our system was studied on these dimpled surfaces for three different spacings. One of these spacings produced first-order like behaviour and was focussed on for much of the thesis. We also investigated how changing the geometry and the inelasticity at the boundary affects the wetting of different phases. This allowed us to spatially control the coexisting liquid and solid phases. Our findings showed behaviour similar to wetting in equilibrium systems. Furthermore, I present a quantitative study confirming the first-order nature of the phase transition in this system. While doing this, I demonstrate evidence of coexistence, hysteresis and surface tension which are all ideas that are commonly associated with first-order phase transitions in equilibrium systems. Inspired by the hydrophobic effect observed in equilibrium systems, a similar effect called the orderphobic effect was recently proposed. This is where disorder inducing intruders placed in an ordered solid experience a force of attraction. The authors suggest that this effect should be general to any system that experiences a first-order order-disorder phase transition. Since our results showed the necessary pre-prerequisites for observing such an effect, we investigated whether such a force could be observed. Although our attempts to reproduce this effect in our non-equilibrium system were inconclusive, we believe the results are promising for future investigation. Finally, I present a more detailed investigation into how changing the spacing of the dimpled lattice changes the nature of the transitions for a broader range of spacings. Our results indicate that different phases form depending on the lattice spacing. We also discuss how the equilibrium ideas of stability can be applied to the system using spacings that display a combination of different phases

    Perceptually Uniform Construction of Illustrative Textures

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    Illustrative textures, such as stippling or hatching, were predominantly used as an alternative to conventional Phong rendering. Recently, the potential of encoding information on surfaces or maps using different densities has also been recognized. This has the significant advantage that additional color can be used as another visual channel and the illustrative textures can then be overlaid. Effectively, it is thus possible to display multiple information, such as two different scalar fields on surfaces simultaneously. In previous work, these textures were manually generated and the choice of density was unempirically determined. Here, we first want to determine and understand the perceptual space of illustrative textures. We chose a succession of simplices with increasing dimensions as primitives for our textures: Dots, lines, and triangles. Thus, we explore the texture types of stippling, hatching, and triangles. We create a range of textures by sampling the density space uniformly. Then, we conduct three perceptual studies in which the participants performed pairwise comparisons for each texture type. We use multidimensional scaling (MDS) to analyze the perceptual spaces per category. The perception of stippling and triangles seems relatively similar. Both are adequately described by a 1D manifold in 2D space. The perceptual space of hatching consists of two main clusters: Crosshatched textures, and textures with only one hatching direction. However, the perception of hatching textures with only one hatching direction is similar to the perception of stippling and triangles. Based on our findings, we construct perceptually uniform illustrative textures. Afterwards, we provide concrete application examples for the constructed textures.Comment: 11 pages, 15 figures, to be published in IEEE Transactions on Visualization and Computer Graphic

    Functional space-time properties of team synergies in high-performance football

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    This thesis aimed to investigate the performance of high-level teams in football, through the analysis of the interactions of their players in the context of the game, as these interactions result in functional effects that could not otherwise be achieved (synergies). From a spatial point of view, we argue that the understanding of collective “payoffs” emerging from players’ interactions and their behavioural patterns, can be accomplished through ”Delaunay triangulations” and consequent ”Voronoi diagrams”. Analysing the positional data (22 players and the ball) in 20 games of the French premier league, in this thesis we essentially sought to focus on territorial dominance as a variable that potentially captures the spatial affordances perceived by players. Whether from a collective global point of view or from a perspective of the local interactions that arise in the game landscape. Supported by the ecological dynamics and the synergism hypothesis, in this thesis we begin by demonstrating the existing connection between the territorial dominance of a team and the offensive effectiveness, as well as the absence of temporal overlap between the ball possession status and territorial dominance. Similarly, we also demonstrated that the space dominance of each player, which contributes to the territorial dominance of the team as a whole, is constrained by the team’s formation and the role assumed by each player in this collective framework. In order to understand the dynamics of interactions between players and the functional effects that come from it, we then focus on two tasks that are related to collective performance: the pass and the shot. Reflecting on the need to find methods that capture how the distribution of players on the pitch influences the functional degrees of freedom of a team as a whole and the passing opportunities that emerge from it. And, at the level of finishing situations, how the dominance of space can be included in the quantification of the value that each player assigns to occupy a certain place in the game landscape, and which is at the basis of their decision-making (shoot or pass the ball to another teammate possibly better ”positioned”). In sum, through the initial conceptual framework and the applied studies, we argue that the analysis of team performance should focus on the functional synergies that result from interactions between players. In this way, we demonstrate, through some examples, how the methods and conclusions taken from this thesis can be applied in practice by football coaches.Esta tese teve como objetivo investigar a performance de equipas de alto nível no futebol, através da análise das interações dos seus jogadores no contexto do jogo pois daí resultam efeitos funcionais que apenas são atingidos através dessas mesmas interações (sinergias). De um ponto de vista espacial, defendemos que o estudo glocal das interações entre os jogadores para a compreensão do rendimento coletivo, pode ser realizado através de “triangulações de Delaunay” e consequentes “diagramas de Voronoi”. Analisando os dados posicionais dos 22 jogadores e da bola, em 20 jogos da primeira liga francesa, nesta tese procurámos essencialmente nos focar sobre o domínio territorial enquanto variável que capta potencialmente as affordances espaciais percebidas pelos jogadores. Seja de um ponto de vista global coletivo, seja numa perspetiva das interações locais que surgem na paisagem de jogo. Suportados pela dinâmica ecológica e pela hipótese do sinergismo, nesta tese começamos por demonstrar a ligação existente entre o domínio territorial das equipas e a sua efetividade ofensiva, bem como a inexistência de uma sobreposição temporal entre a posse de bola e esse domínio. De igual forma, também demonstrámos que o domínio do espaço de cada jogador, que contribui para o domínio territorial da equipa no seu todo, é constrangido pelo sistema de jogo das equipas e pelo papel assumido por cada jogador neste referencial coletivo. No sentido de compreender a dinâmica das interações entre os jogadores e os efeitos funcionais que daí advêm, focamo-nos seguidamente em duas tarefas que estão relacionadas com a performance coletiva: o passe e o remate. Refletindo sobre a necessidade de encontrar métodos que captem de que forma a distribuição dos jogadores em campo influencia os graus de liberdade funcionais de uma equipa no seu todo e as oportunidades de passe que daí emergem. E, ao nível das situações de finalização, de que forma o domínio do espaço poderá ser incluído na quantificação do valor que cada jogador atribui a ocupar um determinador espaço na paisagem de jogo e que está na base da sua tomada de decisão (rematar ou passar a bola para outro colega eventualmente melhor “posicionado”). Em suma, através do enquadramento conceptual inicial e dos estudos aplicados, defendemos que o estudo da performance das equipas deverá se centrar nas sinergias funcionais que resultam das interações entre os jogadores. Desta forma, demonstramos, através de alguns exemplos, como é que os métodos e ilações retirados desta tese poderão ser aplicados na prática pelos treinadores de futebol

    Intrinsic Gaussian process on unknown manifolds with probabilistic metrics

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    This article presents a novel approach to construct Intrinsic Gaussian Processes for regression on unknown manifolds with probabilistic metrics (GPUM ) in point clouds. In many real world applications, one often encounters high dimensional data (e.g.‘point cloud data’) centered around some lower dimensional unknown manifolds. The geometry of manifold is in general different from the usual Euclidean geometry. Naively applying traditional smoothing methods such as Euclidean Gaussian Processes (GPs) to manifold-valued data and so ignoring the geometry of the space can potentially lead to highly misleading predictions and inferences. A manifold embedded in a high dimensional Euclidean space can be well described by a probabilistic mapping function and the corresponding latent space. We investigate the geometrical structure of the unknown manifolds using the Bayesian Gaussian Processes latent variable models(B-GPLVM) and Riemannian geometry. The distribution of the metric tensor is learned using B-GPLVM. The boundary of the resulting manifold is defined based on the uncertainty quantification of the mapping. We use the probabilistic metric tensor to simulate Brownian Motion paths on the unknown manifold. The heat kernel is estimated as the transition density of Brownian Motion and used as the covariance functions of GPUM . The applications of GPUM are illustrated in the simulation studies on the Swiss roll, high dimensional real datasets of WiFi signals and image data examples. Its performance is compared with the Graph Laplacian GP, Graph Mat´ern GP and Euclidean GP

    Drift-diffusion models for innovative semiconductor devices and their numerical solution

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    We present charge transport models for novel semiconductor devices which may include ionic species as well as their thermodynamically consistent finite volume discretization

    Optical and hyperspectral image analysis for image-guided surgery

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    Robust localization and navigation with linear programming

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    Linear programming is an established, well-understood technique optimization problem; the goal of this thesis is to show that we can still use linear programming to advance the state of the art in two important blocks of modern robotic systems, namely perception, and control. In the context of perception, we study the effects of outliers in the solution of localization problems. In its essence, this problem reduces to finding the coordinates of a set of nodes in a common reference frame starting from relative pairwise measurements and is at the core of many applications such as Structure from Motion (SfM), sensor networks, and Simultaneous Localization And Mapping (SLAM). In practical situations, the accuracy of the relative measurements is marred by noise and outliers (large-magnitude errors). In particular, outliers might introduce significant errors in the final result, hence, we have the problem of quantifying how much we should trust the solution returned by some given localization solver. In this work, we focus on the question of whether an L1-norm robust optimization formulation can recover a solution that is identical to the ground truth, under the scenario of translation-only measurements corrupted exclusively by outliers and no noise. In the context of control, we study the problem of robust path planning. Path planning deals with the problem of finding a path from an initial state toward a goal state while considering collision avoidance. We propose a novel approach for navigating in polygonal environments by synthesizing controllers that take as input relative displacement measurements with respect to a set of landmarks. Our algorithm is based on solving a sequence of robust min-max Linear Programming problems on the elements of a cell decomposition of the environment. The optimization problems are formulated using linear Control Lyapunov Function (CLF) and Control Barrier Function (CBF) constraints, to provide stability and safety guarantees, respectively. We integrate the CBF and CLF constraints with sampling-based path planning methods to omit the assumption of having a polygonal environment and add implementation to learn the constraints and estimate the controller when the environment is not fully known. We introduce a method to find the controller synthesis using bearing-only measurements in order to use monocular camera measurements. We show through simulations that the resulting controllers are robust to significant deformations of the environment. These works provide a simple approach in terms of computation to study the robustness of the localization and navigation problem

    Revisiting Random Points: Combinatorial Complexity and Algorithms

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    Consider a set PP of nn points picked uniformly and independently from [0,1]d[0,1]^d for a constant dimension dd -- such a point set is extremely well behaved in many aspects. For example, for a fixed r[0,1]r \in [0,1], we prove a new concentration result on the number of pairs of points of PP at a distance at most rr -- we show that this number lies in an interval that contains only O(nlogn)O(n \log n) numbers. We also present simple linear time algorithms to construct the Delaunay triangulation, Euclidean MST, and the convex hull of the points of PP. The MST algorithm is an interesting divide-and-conquer algorithm which might be of independent interest. We also provide a new proof that the expected complexity of the Delaunay triangulation of PP is linear -- the new proof is simpler and more direct, and might be of independent interest. Finally, we present a simple O~(n4/3)\tilde{O}(n^{4/3}) time algorithm for the distance selection problem for d=2d=2
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