1,482 research outputs found
Dynamic Connectivity in Disk Graphs
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
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 toy models constructed from the Kreuzer-Skarke database
up to the maximum Hodge number . 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
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
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
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
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
We present charge transport models for novel semiconductor devices which may include ionic species as well as their thermodynamically consistent finite volume discretization
Robust localization and navigation with linear programming
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
Consider a set of points picked uniformly and independently from
for a constant dimension -- such a point set is extremely well
behaved in many aspects. For example, for a fixed , we prove a new
concentration result on the number of pairs of points of at a distance at
most -- we show that this number lies in an interval that contains only
numbers.
We also present simple linear time algorithms to construct the Delaunay
triangulation, Euclidean MST, and the convex hull of the points of . 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 is linear -- the new proof is simpler and
more direct, and might be of independent interest. Finally, we present a simple
time algorithm for the distance selection problem for
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