1,075 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Development, Implementation, and Optimization of a Modern, Subsonic/Supersonic Panel Method

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    In the early stages of aircraft design, engineers consider many different design concepts, examining the trade-offs between different component arrangements and sizes, thrust and power requirements, etc. Because so many different designs are considered, it is best in the early stages of design to use simulation tools that are fast; accuracy is secondary. A common simulation tool for early design and analysis is the panel method. Panel methods were first developed in the 1950s and 1960s with the advent of modern computers. Despite being reasonably accurate and very fast, their development was abandoned in the late 1980s in favor of more complex and accurate simulation methods. The panel methods developed in the 1980s are still in use by aircraft designers today because of their accuracy and speed. However, they are cumbersome to use and limited in applicability. The purpose of this work is to reexamine panel methods in a modern context. In particular, this work focuses on the application of panel methods to supersonic aircraft (a supersonic aircraft is one that flies faster than the speed of sound). Various aspects of the panel method, including the distributions of the unknown flow variables on the surface of the aircraft and efficiently solving for these unknowns, are discussed. Trade-offs between alternative formulations are examined and recommendations given. This work also serves to bring together, clarify, and condense much of the literature previously published regarding panel methods so as to assist future developers of panel methods

    Introduction to Riemannian Geometry and Geometric Statistics: from basic theory to implementation with Geomstats

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    International audienceAs data is a predominant resource in applications, Riemannian geometry is a natural framework to model and unify complex nonlinear sources of data.However, the development of computational tools from the basic theory of Riemannian geometry is laborious.The work presented here forms one of the main contributions to the open-source project geomstats, that consists in a Python package providing efficient implementations of the concepts of Riemannian geometry and geometric statistics, both for mathematicians and for applied scientists for whom most of the difficulties are hidden under high-level functions. The goal of this monograph is two-fold. First, we aim at giving a self-contained exposition of the basic concepts of Riemannian geometry, providing illustrations and examples at each step and adopting a computational point of view. The second goal is to demonstrate how these concepts are implemented in Geomstats, explaining the choices that were made and the conventions chosen. The general concepts are exposed and specific examples are detailed along the text.The culmination of this implementation is to be able to perform statistics and machine learning on manifolds, with as few lines of codes as in the wide-spread machine learning tool scikit-learn. We exemplify this with an introduction to geometric statistics

    Understanding X-ray Pulsars: from Blind Source Separation to Pulse Profile Decompositions

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    In Röntgendoppelsternen mit einem Neutronenstern wird Materie vom Begleiter auf das kompakte Objekt übertragen, wo sie auf das starke Magnetfeld der Magnetosphäre trifft. Das Magnetfeld lenkt die Materie zu den magnetischen Polen, wo sie ihre Gravitationsenergie abgibt, hauptsächlich in Form von Röntgenstrahlung. Wenn die Spin- und Magnetachse versetzt sind, kann diese Emission als Pulsation beobachtet werden, und das Objekt wird dann als akkretierender Röntgenpulsar bezeichnet. Faltet man die Lichtkurve, d. h. die Intensitätsvariation der Emission über die Zeit, mit der Spinperiode des Pulsars, erhält man ein Pulsprofil, das aufgrund des Zusammenspiels verschiedener Faktoren wie Geometrie, Magnetfeldkonfiguration, intrinsische Strahlenmuster und gravitationsbedingte Lichtablenkung komplexe Formen aufweisen kann. Die Physik von Röntgenpulsaren unter ihren extremen Bedingungen zu verstehen, stellt eine Reihe von Herausforderungen dar. Der Strahlungstransport und die dynamische Struktur des Akkretionsflusses müssen modelliert werden, um ihre komplexe Natur zu verstehen. Diese Probleme sind miteinander verknüpft, da die Eigenschaften und die Dynamik der akkretierenden Materie die Röntgenemission direkt beeinflussen. Diese Wechselwirkungen spielen eine wichtige Rolle bei der Entstehung der beobachteten Röntgenspektren. Sie zu verstehen ist sowohl aus theoretischer als auch aus beobachtungstechnischer Sicht keine triviale Aufgabe. Hinzu kommt, dass die Emission aus der unmittelbaren Umgebung der Neutronensternoberfläche aufgrund von Phänomenen wie der gravitationsbedingten Lichtablenkung nicht unbedingt mit der beobachteten Strahlung übereinstimmt. Außerdem kann während verschiedener Phasen der Sternrotation die Emission von beiden Polen gleichzeitig beobachtet werden. Um Modelle zu testen und zu verfeinern, die die Entstehung von Spektren und Pulsprofilen in Röntgenpulsaren beschreiben, müssen die Beiträge beider Pole zum beobachteten Fluss als Funktion der Phase bestimmt werden. Dies ermöglicht die Untersuchung der physikalischen Prozesse, die am Strahlungstransport beteiligt sind. Die Rotation des Pulsars erlaubt es, die Winkelabhängigkeit der Emission zu untersuchen, was wiederum zu einem besseren Verständnis der Physik von Röntgenpulsaren beiträgt. In dieser Arbeit wird ein neuer datenbasierter Ansatz verwendet, der die Puls-zu-Puls-Variabilität des beobachteten Flusses ausnutzt. Die Methode basiert auf der Behandlung der Aufgabe als ein Problem der blinden Quellentrennung. Das Ziel ist die Schätzung unbekannter Signale, die mit unbekannten Mischkoeffizienten gemischt sind. Dabei kann die inhärente Variabilität des Flusses genutzt werden, die teilweise unabhängig von den beiden Polen ist. Das Ergebnis sind zwei Signale, die die Flussvariabilität jedes Pols darstellen, skaliert durch die beiden Gewichtungen, die jedem Pol zugeordnet sind. Diese Gewichtungen sind die einpoligen Pulsprofile, die von Interesse sind. Um die Methode zu etablieren, entwickle ich in dieser Arbeit die phasenkorrelierte Variabilitätsanalyse (PCVA) durch eine Reihe von Simulationen. Dies beinhaltet die Bestimmung der Anforderungen und Grenzen der PCVA, um ihre Effektivität bei der Trennung der Beiträge beider Pole in Röntgenpulsaren zu bestimmen. Anschließend demonstriere ich die Anwendung der PCVA auf Beobachtungsdaten des hellen Röntgenpulsars Cen X-3, die aus RXTE (Rossi X-ray Timing Explorer) Beobachtungen stammen. Ich vergleiche die Ergebnisse der PCVA mit früheren Arbeiten, die sich mit dem gleichen Problem beschäftigten. Auf der Grundlage meiner Ergebnisse stelle ich fest, dass die in der Vergangenheit getroffene Annahme der Symmetrie der intrinsischen Strahlungsmuster mit den Ergebnissen der PCVA unvereinbar und daher möglicherweise nicht gerechtfertigt ist. Um die Ergebnisse der PCVA interpretieren zu können, habe ich ein einfaches Modell zur Beschreibung der erhaltenen Pulsprofile erstellt. Dieses Modell basiert auf einer Reihe von Annahmen. Zunächst wird die Geometrie des Systems unter Berücksichtigung der Inklination und des Positionswinkels des Pulsarspins definiert. Es wird angenommen, dass die Emissionsregion von einer einzelnen Quelle an jedem Pol erzeugt wird. Um eine Asymmetrie einzuführen, wird ein phänomenologisches Strahlungsmuster definiert, das symmetrisch zu einer bestimmten Richtung, aber asymmetrisch zur Oberflächennormalen ist. Dies führt zu asymmetrischen Strahlenmustern aufgrund der Rotation des Neutronensterns. Das Modell ermöglicht es auch eine Dipolverschiebung als zusätzlichen Parameter zu berücksichtigen und die Effekte der gravitationsbedingten Lichtablenkung miteinzubeziehen. Auf diese Weise bietet das Modell die Möglichkeit den Einfluss verschiedener Parameter, insbesondere des intrinsischen Strahlungsmusters, auf die beobachteten Pulsprofile zu untersuchen. Ich verwende das Modell, um das Cen X-3 PCVA-Ergebnis zu untersuchen. Da die meisten Beiträge während der Rotation nicht Null sind, ist es naheliegend, dass die Emissionsregionen fast immer sichtbar sind. Trotz der Berücksichtigung der gravitationsbedingten Lichtablenkung ist eine Lösung, die dies widerspiegelt, auf der Basis der Literaturgeometrie schwierig. Eine leichte Verbesserung wird durch die Einbeziehung einer Dipolverschiebung erreicht, aber das Problem der fehlenden Sichtbarkeit der Emissionsregion bleibt bestehen. Ich diskutiere mögliche Lösungen für dieses Problem. Eine Möglichkeit besteht darin, die Beschränkung der Pulsargeometrie zu lockern, da alle Methoden zur Bestimmung der Geometrie auf Modellen oder Annahmen beruhen, die nicht absolut sind. Eine andere Möglichkeit wäre, ein komplexeres Strahlungsmodell zu verwenden oder ausgedehntere Emissionsregionen zu berücksichtigen. Die in dieser Arbeit entwickelte PCVA erlaubt die Trennung der Emissionsbeiträge der beiden Pole von Röntgenpulsaren. Diese Methode kann prinzipiell auf jeden leuchtkräftigen Röntgenpulsar angewendet werden, sofern die spezifizierten Quellen- und Beobachtungsbedingungen erfüllt sind. Durch die Untersuchung der individuellen Beiträge der beiden Pole ermöglicht die PCVA die Untersuchung von Veränderungen im Akkretionsprozess und in den Akkretionsstrukturen. Darüber hinaus bietet das von mir entwickelte Modell ein Werkzeug zur Untersuchung verschiedener Parameter die das Pulsprofil formen, insbesondere die Auswirkungen eines leicht asymmetrischen Strahlungsmusters.In neutron star X-ray binaries, matter is transferred from the companion to the compact object where it encounters the strong magnetic field at the magnetosphere. The magnetic field redirects the material toward the magnetic poles, where it releases its gravitational potential energy mainly as X-rays. If the spin and magnetic axes are misaligned, this emission can be observed as pulsations, and the object is called an accreting X-ray pulsar. Folding the light curve, i.e. the intensity variation of the emission over time, with the spin period of the pulsar gives a pulse profile, which can have complex shapes due to the interplay of several factors such as geometry, magnetic field configuration, intrinsic beam patterns, and gravitational light bending. Understanding the physics of X-ray pulsars, and the extreme conditions that are associated with them, presents a number of challenges. Radiative transfer and the dynamical structure of the accretion flow must be modeled to gain insight into their complex nature. These problems are interrelated, since the properties and dynamics of the accreting matter directly affect the X-ray emission, and the resulting radiation pressure in turn changes the accretion flow. Such interactions play an important role in shaping the observed X-ray spectra and they are difficult to understand from both a theoretical and observational point of view. In addition, the emission that escapes from the immediate vicinity of the surface of the neutron star is not necessarily what is observed: during different phases of the star's rotation, emission from both poles may be observed simultaneously due to phenomena such as gravitational light bending. To test and refine models describing the formation of spectra and pulse profiles in X-ray pulsars, it is necessary to determine the contributions of each pole to the observed flux as a function of phase, which helps to study the physical processes involved in radiative transfer. The rotation of the pulsar makes it possible to study the angular dependence of the emission, which in turn contributes to a better understanding of the physics of X-ray pulsars. To accomplish this, a new data-driven approach is used in this thesis that takes advantage of the pulse-to-pulse variability in the observed flux. The method is based on treating the task as a blind source separation problem, where the goal is to estimate unknown signals mixed with unknown mixing coefficients. In this context, blind source separation techniques can be used by exploiting the inherent flux variability that is partially independent for the two poles. The result of this decomposition are the two signals representing the time variability of the accretion rate at each pole and the weights associated with each pole. These weights are the single-pole pulse profiles of interest. To establish the method, in this thesis I develop the phase correlated variability analysis (PCVA) through a series of simulations. This includes determining the requirements and limitations of the PCVA to ensure its effectiveness in disentangling the contributions of the two poles in X-ray pulsars. I then demonstrate the application of the PCVA to observational data of the bright persistent X-ray pulsar Cen X-3 obtained from RXTE (Rossi X-ray Timing Explorer) observations. I compare the results obtained with the PCVA to those of previous studies that have addressed the same problem. Based on my results, I find that the symmetry assumption made in the past is incompatible with the PCVA results and thus may not be justified. In order to interpret the results of the PCVA, I create a toy model to describe the obtained pulse profiles. This model is based on a number of assumptions. First, the geometry of the system is defined by considering the inclination and position angle of the pulsar spin. The emission region is assumed to be generated by a single source at each pole. To introduce asymmetry, a phenomenological beam pattern is defined that is symmetric about a certain direction but asymmetric with respect to the normal of the surface. This results in asymmetric beam patterns due to the rotation of the neutron star. The toy model also allows for an offset of the dipole as an additional parameter and incorporates the effects of gravitational light bending. In this way, the model provides a means to study the effect of different parameters, most importantly the intrinsic beam pattern, on the observed pulse profiles from each pole. I then address the Cen X-3 PCVA result in the context of the toy model. Despite accounting for gravitational light bending, the presence of mostly non-zero contributions throughout the rotation makes it difficult to find a solution using literature geometry. I find a slight improvement by allowing for an offset of the dipole, but the visibility problem still remains and I discuss possible solutions to this problem. One possibility is to relax the basic pulsar geometry, since all methods for determining the geometry are based on models or assumptions that are not absolute. Alternatively, the use of a more complex beam pattern model or the consideration of extended emission regions could potentially resolve the lack of visibility gaps in the results. The PCVA developed in this thesis allows the separation of the emission contributions from the two poles of X-ray pulsars. In principle, this method can be applied to any luminous X-ray pulsar, provided that the specified source and observational requirements are met. Thus, by studying the individual contributions from each pole, the PCVA allows the study of changes in the accretion process and structures. In addition, the toy model provides a tool for exploring various parameters that shape the pulse profile, in particular the effects of a slightly asymmetric beam pattern

    Euler Characteristic Tools For Topological Data Analysis

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    In this article, we study Euler characteristic techniques in topological data analysis. Pointwise computing the Euler characteristic of a family of simplicial complexes built from data gives rise to the so-called Euler characteristic profile. We show that this simple descriptor achieve state-of-the-art performance in supervised tasks at a very low computational cost. Inspired by signal analysis, we compute hybrid transforms of Euler characteristic profiles. These integral transforms mix Euler characteristic techniques with Lebesgue integration to provide highly efficient compressors of topological signals. As a consequence, they show remarkable performances in unsupervised settings. On the qualitative side, we provide numerous heuristics on the topological and geometric information captured by Euler profiles and their hybrid transforms. Finally, we prove stability results for these descriptors as well as asymptotic guarantees in random settings.Comment: 39 page

    Aerial Drone-based System for Wildfire Monitoring and Suppression

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    Wildfire, also known as forest fire or bushfire, being an uncontrolled fire crossing an area of combustible vegetation, has become an inherent natural feature of the landscape in many regions of the world. From local to global scales, wildfire has caused substantial social, economic and environmental consequences. Given the hazardous nature of wildfire, developing automated and safe means to monitor and fight the wildfire is of special interest. Unmanned aerial vehicles (UAVs), equipped with appropriate sensors and fire retardants, are available to remotely monitor and fight the area undergoing wildfires, thus helping fire brigades in mitigating the influence of wildfires. This thesis is dedicated to utilizing UAVs to provide automated surveillance, tracking and fire suppression services on an active wildfire event. Considering the requirement of collecting the latest information of a region prone to wildfires, we presented a strategy to deploy the estimated minimum number of UAVs over the target space with nonuniform importance, such that they can persistently monitor the target space to provide a complete area coverage whilst keeping a desired frequency of visits to areas of interest within a predefined time period. Considering the existence of occlusions on partial segments of the sensed wildfire boundary, we processed both contour and flame surface features of wildfires with a proposed numerical algorithm to quickly estimate the occluded wildfire boundary. To provide real-time situational awareness of the propagated wildfire boundary, according to the prior knowledge of the whole wildfire boundary is available or not, we used the principle of vector field to design a model-based guidance law and a model-free guidance law. The former is derived from the radial basis function approximated wildfire boundary while the later is based on the distance between the UAV and the sensed wildfire boundary. Both vector field based guidance laws can drive the UAV to converge to and patrol along the dynamic wildfire boundary. To effectively mitigate the impacts of wildfires, we analyzed the advancement based activeness of the wildfire boundary with a signal prominence based algorithm, and designed a preferential firefighting strategy to guide the UAV to suppress fires along the highly active segments of the wildfire boundary

    Unifying O(3) Equivariant Neural Networks Design with Tensor-Network Formalism

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    Many learning tasks, including learning potential energy surfaces from ab initio calculations, involve global spatial symmetries and permutational symmetry between atoms or general particles. Equivariant graph neural networks are a standard approach to such problems, with one of the most successful methods employing tensor products between various tensors that transform under the spatial group. However, as the number of different tensors and the complexity of relationships between them increase, maintaining parsimony and equivariance becomes increasingly challenging. In this paper, we propose using fusion diagrams, a technique widely employed in simulating SU(22)-symmetric quantum many-body problems, to design new equivariant components for equivariant neural networks. This results in a diagrammatic approach to constructing novel neural network architectures. When applied to particles within a given local neighborhood, the resulting components, which we term "fusion blocks," serve as universal approximators of any continuous equivariant function defined in the neighborhood. We incorporate a fusion block into pre-existing equivariant architectures (Cormorant and MACE), leading to improved performance with fewer parameters on a range of challenging chemical problems. Furthermore, we apply group-equivariant neural networks to study non-adiabatic molecular dynamics of stilbene cis-trans isomerization. Our approach, which combines tensor networks with equivariant neural networks, suggests a potentially fruitful direction for designing more expressive equivariant neural networks.Comment: 10 pages + 12-page supplementary materials, many figure

    Geometric optimization problems in quantum computation and discrete mathematics: Stabilizer states and lattices

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    This thesis consists of two parts: Part I deals with properties of stabilizer states and their convex hull, the stabilizer polytope. Stabilizer states, Pauli measurements and Clifford unitaries are the three building blocks of the stabilizer formalism whose computational power is limited by the Gottesman- Knill theorem. This model is usually enriched by a magic state to get a universal model for quantum computation, referred to as quantum computation with magic states (QCM). The first part of this thesis will investigate the role of stabilizer states within QCM from three different angles. The first considered quantity is the stabilizer extent, which provides a tool to measure the non-stabilizerness or magic of a quantum state. It assigns a quantity to each state roughly measuring how many stabilizer states are required to approximate the state. It has been shown that the extent is multiplicative under taking tensor products when the considered state is a product state whose components are composed of maximally three qubits. In Chapter 2, we will prove that this property does not hold in general, more precisely, that the stabilizer extent is strictly submultiplicative. We obtain this result as a consequence of rather general properties of stabilizer states. Informally our result implies that one should not expect a dictionary to be multiplicative under taking tensor products whenever the dictionary size grows subexponentially in the dimension. In Chapter 3, we consider QCM from a resource theoretic perspective. The resource theory of magic is based on two types of quantum channels, completely stabilizer preserving maps and stabilizer operations. Both classes have the property that they cannot generate additional magic resources. We will show that these two classes of quantum channels do not coincide, specifically, that stabilizer operations are a strict subset of the set of completely stabilizer preserving channels. This might have the consequence that certain tasks which are usually realized by stabilizer operations could in principle be performed better by completely stabilizer preserving maps. In Chapter 4, the last one of Part I, we consider QCM via the polar dual stabilizer polytope (also called the Lambda-polytope). This polytope is a superset of the quantum state space and every quantum state can be written as a convex combination of its vertices. A way to classically simulate quantum computing with magic states is based on simulating Pauli measurements and Clifford unitaries on the vertices of the  Lambda-polytope. The complexity of classical simulation with respect to the polytope   is determined by classically simulating the updates of vertices under Clifford unitaries and Pauli measurements. However, a complete description of this polytope as a convex hull of its vertices is only known in low dimensions (for up to two qubits or one qudit when odd dimensional systems are considered). We make progress on this question by characterizing a certain class of operators that live on the boundary of the  Lambda-polytope when the underlying dimension is an odd prime. This class encompasses for instance Wigner operators, which have been shown to be vertices of  Lambda. We conjecture that this class contains even more vertices of  Lambda. Eventually, we will shortly sketch why applying Clifford unitaries and Pauli measurements to this class of operators can be efficiently classically simulated. Part II of this thesis deals with lattices. Lattices are discrete subgroups of the Euclidean space. They occur in various different areas of mathematics, physics and computer science. We will investigate two types of optimization problems related to lattices. In Chapter 6 we are concerned with optimization within the space of lattices. That is, we want to compare the Gaussian potential energy of different lattices. To make the energy of lattices comparable we focus on lattices with point density one. In particular, we focus on even unimodular lattices and show that, up to dimension 24, they are all critical for the Gaussian potential energy. Furthermore, we find that all n-dimensional even unimodular lattices with n   24 are local minima or saddle points. In contrast in dimension 32, there are even unimodular lattices which are local maxima and others which are not even critical. In Chapter 7 we consider flat tori R^n/L, where L is an n-dimensional lattice. A flat torus comes with a metric and our goal is to approximate this metric with a Hilbert space metric. To achieve this, we derive an infinite-dimensional semidefinite optimization program that computes the least distortion embedding of the metric space R^n/L into a Hilbert space. This program allows us to make several interesting statements about the nature of least distortion embeddings of flat tori. In particular, we give a simple proof for a lower bound which gives a constant factor improvement over the previously best lower bound on the minimal distortion of an embedding of an n-dimensional flat torus. Furthermore, we show that there is always an optimal embedding into a finite-dimensional Hilbert space. Finally, we construct optimal least distortion embeddings for the standard torus R^n/Z^n and all 2-dimensional flat tori

    Data analysis with merge trees

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    Today’s data are increasingly complex and classical statistical techniques need growingly more refined mathematical tools to be able to model and investigate them. Paradigmatic situations are represented by data which need to be considered up to some kind of trans- formation and all those circumstances in which the analyst finds himself in the need of defining a general concept of shape. Topological Data Analysis (TDA) is a field which is fundamentally contributing to such challenges by extracting topological information from data with a plethora of interpretable and computationally accessible pipelines. We con- tribute to this field by developing a series of novel tools, techniques and applications to work with a particular topological summary called merge tree. To analyze sets of merge trees we introduce a novel metric structure along with an algorithm to compute it, define a framework to compare different functions defined on merge trees and investigate the metric space obtained with the aforementioned metric. Different geometric and topolog- ical properties of the space of merge trees are established, with the aim of obtaining a deeper understanding of such trees. To showcase the effectiveness of the proposed metric, we develop an application in the field of Functional Data Analysis, working with functions up to homeomorphic reparametrization, and in the field of radiomics, where each patient is represented via a clustering dendrogram
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