19,608 research outputs found

    Multi-Objective Trust-Region Filter Method for Nonlinear Constraints using Inexact Gradients

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    In this article, we build on previous work to present an optimization algorithm for nonlinearly constrained multi-objective optimization problems. The algorithm combines a surrogate-assisted derivative-free trust-region approach with the filter method known from single-objective optimization. Instead of the true objective and constraint functions, so-called fully linear models are employed, and we show how to deal with the gradient inexactness in the composite step setting, adapted from single-objective optimization as well. Under standard assumptions, we prove convergence of a subset of iterates to a quasi-stationary point and if constraint qualifications hold, then the limit point is also a KKT-point of the multi-objective problem

    On questions of uniqueness for the vacant set of Wiener sausages and Brownian interlacements

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    We consider connectivity properties of the vacant set of (random) ensembles of Wiener sausages in Rd\mathbb R^d in the transient dimensions d≥3d \geq 3. We prove that the vacant set of Brownian interlacements contains at most one infinite connected component almost surely. For finite ensembles of Wiener sausages, we provide sharp polynomial bounds on the probability that their vacant set contains at least 22 connected components in microscopic balls. The main proof ingredient is a sharp polynomial bound on the probability that several Brownian motions visit jointly all hemiballs of the unit ball while avoiding a slightly smaller ball.Comment: 40 page

    The Metaverse: Survey, Trends, Novel Pipeline Ecosystem & Future Directions

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    The Metaverse offers a second world beyond reality, where boundaries are non-existent, and possibilities are endless through engagement and immersive experiences using the virtual reality (VR) technology. Many disciplines can benefit from the advancement of the Metaverse when accurately developed, including the fields of technology, gaming, education, art, and culture. Nevertheless, developing the Metaverse environment to its full potential is an ambiguous task that needs proper guidance and directions. Existing surveys on the Metaverse focus only on a specific aspect and discipline of the Metaverse and lack a holistic view of the entire process. To this end, a more holistic, multi-disciplinary, in-depth, and academic and industry-oriented review is required to provide a thorough study of the Metaverse development pipeline. To address these issues, we present in this survey a novel multi-layered pipeline ecosystem composed of (1) the Metaverse computing, networking, communications and hardware infrastructure, (2) environment digitization, and (3) user interactions. For every layer, we discuss the components that detail the steps of its development. Also, for each of these components, we examine the impact of a set of enabling technologies and empowering domains (e.g., Artificial Intelligence, Security & Privacy, Blockchain, Business, Ethics, and Social) on its advancement. In addition, we explain the importance of these technologies to support decentralization, interoperability, user experiences, interactions, and monetization. Our presented study highlights the existing challenges for each component, followed by research directions and potential solutions. To the best of our knowledge, this survey is the most comprehensive and allows users, scholars, and entrepreneurs to get an in-depth understanding of the Metaverse ecosystem to find their opportunities and potentials for contribution

    A hybrid quantum algorithm to detect conical intersections

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    Conical intersections are topologically protected crossings between the potential energy surfaces of a molecular Hamiltonian, known to play an important role in chemical processes such as photoisomerization and non-radiative relaxation. They are characterized by a non-zero Berry phase, which is a topological invariant defined on a closed path in atomic coordinate space, taking the value π\pi when the path encircles the intersection manifold. In this work, we show that for real molecular Hamiltonians, the Berry phase can be obtained by tracing a local optimum of a variational ansatz along the chosen path and estimating the overlap between the initial and final state with a control-free Hadamard test. Moreover, by discretizing the path into NN points, we can use NN single Newton-Raphson steps to update our state non-variationally. Finally, since the Berry phase can only take two discrete values (0 or π\pi), our procedure succeeds even for a cumulative error bounded by a constant; this allows us to bound the total sampling cost and to readily verify the success of the procedure. We demonstrate numerically the application of our algorithm on small toy models of the formaldimine molecule (\ce{H2C=NH}).Comment: 15 + 10 pages, 4 figure

    Adaptive measurement filter: efficient strategy for optimal estimation of quantum Markov chains

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    Continuous-time measurements are instrumental for a multitude of tasks in quantum engineering and quantum control, including the estimation of dynamical parameters of open quantum systems monitored through the environment. However, such measurements do not extract the maximum amount of information available in the output state, so finding alternative optimal measurement strategies is a major open problem. In this paper we solve this problem in the setting of discrete-time input-output quantum Markov chains. We present an efficient algorithm for optimal estimation of one-dimensional dynamical parameters which consists of an iterative procedure for updating a `measurement filter' operator and determining successive measurement bases for the output units. A key ingredient of the scheme is the use of a coherent quantum absorber as a way to post-process the output after the interaction with the system. This is designed adaptively such that the joint system and absorber stationary state is pure at a reference parameter value. The scheme offers an exciting prospect for optimal continuous-time adaptive measurements, but more work is needed to find realistic practical implementations.Comment: 25 pages 7 figure

    Chiral active fluids: Odd viscosity, active turbulence, and directed flows of hydrodynamic microrotors

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    While the number of publications on rotating active matter has rapidly increased in recent years, studies on purely hydrodynamically interacting rotors on the microscale are still rare, especially from the perspective of particle based hydrodynamic simulations. The work presented here targets to fill this gap. By means of high-performance computer simulations, performed in a highly parallelised fashion on graphics processing units, the dynamics of ensembles of up to 70,000 rotating colloids immersed in an explicit mesoscopic solvent consisting out of up to 30 million fluid particles, are investigated. Some of the results presented in this thesis have been worked out in collaboration with experimentalists, such that the theoretical considerations developed in this thesis are supported by experiments, and vice versa. The studied system, modelled in order to resemble the essential physics of the experimentally realisable system, consists out of rotating magnetic colloidal particles, i.e., (micro-)rotors, rotating in sync to an externally applied magnetic field, where the rotors solely interact via hydrodynamic and steric interactions. Overall, the agreement between simulations and experiments is very good, proving that hydrodynamic interactions play a key role in this and related systems. While already an isolated rotating colloid is driven out of equilibrium, only collections of two or more rotors have experimentally shown to be able to convert the rotational energy input into translational dynamics in an orbital rotating fashion. The rotating colloids inject circular flows into the fluid, such that detailed balance is broken, and it is not a priori known whether equilibrium properties of colloids can be extended to isolated rotating colloids. A joint theoretical and experimental analysis of isolated, pairs, and small groups of hydrodynamically interacting rotors is given in chapter 2. While the translational dynamics of isolated rotors effectively resemble the dynamics of non-rotating colloids, the orbital rotation of pairs of rotors can be described with leading order hydrodynamics and a two-dimensional analogy of Faxén’s law is derived. In chapter 3, a homogeneously distributed ensemble of rotors (bulk) as a realisation of a chiral active fluid is studied and it is explicitly shown computationally and experimentally that it carries odd viscosity. The mutual orbital translation of rotors and an increase of the effective solvent viscosity with rotor density lead to a non-monotonous behaviour of the average translational velocity. Meanwhile, the rotor suspension bears a finite osmotic compressibility resulting from the long-ranged nature of hydrody- namic interactions such that rotational and odd stresses are transmitted through the solvent also at small and intermediate rotor densities. Consequently, density inhomogeneities predicted for chiral active fluids with odd viscosity can be found and allow for an explicit measurement of odd viscosity in simulations and experiments. At intermediate densities, the collective dynamics shows the emergence of multi-scale vortices and chaotic motion which is identified as active turbulence with a self-similar power-law decay in the energy spectrum, showing that the injected energy on the rotor scale is transported to larger scales, similar to the inverse energy cascade of clas- sical two-dimensional turbulence. While either odd viscosity or active turbulence have been reported in chiral active matter previously, the system studied here shows that the emergence of both simultaneously is possible resulting from the osmotic compressibility and hydrodynamic mediation of odd and active stresses. The collective dynamics of colloids rotating out of phase, i.e., where a constant torque instead of a constant angular velocity is applied, is shown to be qualitatively very similar. However, at smaller densities, local density inhomogeneities imply position dependent angular velocities of the rotors resulting from inter-rotor friction. While the friction of a quasi-2D layer of active colloids with the substrate is often not easily modifiable in experiments, the incorporation of substrate friction into the simulation models typically implies a considerable increase in computational effort. In chapter 4, a very efficient way of incorporating the friction with a substrate into a two-dimensional multiparticle collision dynamics solvent is introduced, allowing for an explicit investigation of the influences of substrate on active dynamics. For the rotor fluid, it is explicitly shown that the influence of the substrate friction results in a cutoff of the hydrodynamic interaction length, such that the maximum size of the formed vortices is controlled by the substrate friction, also resulting in a cutoff in the energy spectrum, because energy is taken out of the system at the respective length. These findings are in agreement with the experiments. Since active particles in confinement are known to organise in states of collective dynamics, ensembles of rotationally actuated colloids are studied in circular confinement and in the presence of periodic obstacle lattices in chapters 5 and 6, respectively. The results show that the chaotic active turbulent transport of rotors in suspension can be enhanced and guided resulting from edge flows generated at the boundaries, as has recently been reported for a related chiral active system. The consequent collective rotor dynamics can be regarded as a superposition of active turbulent and imposed flows, leading to on average stationary flows. In contrast to the bulk dynamics, the imposed flows inject additional energy into the system on the long length scales, and the same scaling behaviour of the energy spectrum as in bulk is only obtained if the energy injection scales, due to the mutual generation of rotor translational dynamics throughout the system and the edge flows, are well separated. The combination of edge flow and entropic layering at the boundaries leads to oscillating hydrodynamic stresses and consequently to an oscillating vorticity profile. In the presence of odd viscosity, this consequently leads to non-trivial steady-state density modulations at the boundary, resulting from a balance of osmotic pressure and odd stresses. Relevant for the efficient dispersion and mixing of inert particles on the mesoscale by means of active turbulent mixing powered by rotors, a study of the dynamics of a binary mixture consisting out of rotors and passive particles is presented in chapter 7. Because the rotors are not self-propelled, but the translational dynamics is induced by the surrounding rotors, the passive particles, which do not inject further energy into the system, are transported according to the same mechanism as the rotors. The collective dynamics thus resembles the pure rotor bulk dynamics at the respective density of only rotors. However, since no odd stresses act between the passive particles, only mutual rotor interactions lead to odd stresses leading to the accumulation of rotors in the regions of positive vorticity. This density increase is associated with a pressure increase, which balances the odd stresses acting on the rotors. However, the passive particles are only subject to the accumulation induced pressure increase such that these particles are transported into the areas of low rotor concentration, i.e., the regions of negative vorticity. Under conditions of sustained vortex flow, this results in segregation of both particle types. Since local symmetry breaking can convert injected rotational into translational energy, microswimmers can be constructed out of rotor materials when a suitable breaking of symmetry is kept in the vicinity of a rotor. One hypothetical realisation, i.e., a coupled rotor pair consisting out of two rotors of opposite angular velocity and of fixed distance, termed a birotor, are studied in chapter 8. The birotor pumps the fluid into one direction and consequently translates into the opposite direction, and creates a flow field reminiscent of a source doublet, or sliplet flow field. Fixed in space the birotor might be an interesting realisation of a microfluidic pump. The trans- lational dynamics of a birotor can be mapped onto the active Brownian particle model for single swimmers. However, due to the hydrodynamic interactions among the rotors, the birotor ensemble dynamics do not show the emergence of stable motility induced clustering. The reason for this is the flow created by birotor in small aggregates which effectively pushes further arriving birotors away from small aggregates, which eventually are all dispersed by thermal fluctuations

    Richness of dynamics and global bifurcations in systems with a homoclinic figure-eight

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    We consider 2D flows with a homoclinic figure-eight to a dissipative saddle. We study the rich dynamics that such a system exhibits under a periodic forcing. First, we derive the bifurcation diagram using topological techniques. In particular, there is a homoclinic zone in the parameter space with a non-smooth boundary. We provide a complete explanation of this phenomenon relating it to primary quadratic homoclinic tangency curves which end up at some cubic tangency (cusp) points. We also describe the possible attractors that exist (and may coexist) in the system. A main goal of this work is to show how the previous qualitative description can be complemented with quantitative global information. To this end, we introduce a return map model which can be seen as the simplest one which is 'universal' in some sense. We carry out several numerical experiments on the model, to check that all the objects predicted to exist by the theory are found in the model, and also to investigate new properties of the system

    Path integrals and stochastic calculus

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    Path integrals are a ubiquitous tool in theoretical physics. However, their use is sometimes hindered by the lack of control on various manipulations -- such as performing a change of the integration path -- one would like to carry out in the light-hearted fashion that physicists enjoy. Similar issues arise in the field of stochastic calculus, which we review to prepare the ground for a proper construction of path integrals. At the level of path integration, and in arbitrary space dimension, we not only report on existing Riemannian geometry-based approaches that render path integrals amenable to the standard rules of calculus, but also bring forth new routes, based on a fully time-discretized approach, that achieve the same goal. We illustrate these various definitions of path integration on simple examples such as the diffusion of a particle on a sphere.Comment: 96 pages, 4 figures. New title, expanded introduction and additional references. Version accepted in Advandes in Physic

    Explicit spectral gap for Schottky subgroups of SL(2,Z)\mathrm{SL} (2,\mathbb{Z})

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    Let Γ\Gamma be a Schottky subgroup of SL(2,Z)\mathrm{SL} (2,\mathbb{Z}). We establish a uniform and explicit lower bound of the second eigenvalue of the Laplace-Beltrami operator of congruence coverings of the hyperbolic surface Γ\H2\Gamma \backslash \mathbb{H}^2 provided the limit set of Γ\Gamma is thick enough.Comment: 31 page

    Modelling uncertainties for measurements of the H → γγ Channel with the ATLAS Detector at the LHC

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    The Higgs boson to diphoton (H → γγ) branching ratio is only 0.227 %, but this final state has yielded some of the most precise measurements of the particle. As measurements of the Higgs boson become increasingly precise, greater import is placed on the factors that constitute the uncertainty. Reducing the effects of these uncertainties requires an understanding of their causes. The research presented in this thesis aims to illuminate how uncertainties on simulation modelling are determined and proffers novel techniques in deriving them. The upgrade of the FastCaloSim tool is described, used for simulating events in the ATLAS calorimeter at a rate far exceeding the nominal detector simulation, Geant4. The integration of a method that allows the toolbox to emulate the accordion geometry of the liquid argon calorimeters is detailed. This tool allows for the production of larger samples while using significantly fewer computing resources. A measurement of the total Higgs boson production cross-section multiplied by the diphoton branching ratio (σ × Bγγ) is presented, where this value was determined to be (σ × Bγγ)obs = 127 ± 7 (stat.) ± 7 (syst.) fb, within agreement with the Standard Model prediction. The signal and background shape modelling is described, and the contribution of the background modelling uncertainty to the total uncertainty ranges from 18–2.4 %, depending on the Higgs boson production mechanism. A method for estimating the number of events in a Monte Carlo background sample required to model the shape is detailed. It was found that the size of the nominal γγ background events sample required a multiplicative increase by a factor of 3.60 to adequately model the background with a confidence level of 68 %, or a factor of 7.20 for a confidence level of 95 %. Based on this estimate, 0.5 billion additional simulated events were produced, substantially reducing the background modelling uncertainty. A technique is detailed for emulating the effects of Monte Carlo event generator differences using multivariate reweighting. The technique is used to estimate the event generator uncertainty on the signal modelling of tHqb events, improving the reliability of estimating the tHqb production cross-section. Then this multivariate reweighting technique is used to estimate the generator modelling uncertainties on background V γγ samples for the first time. The estimated uncertainties were found to be covered by the currently assumed background modelling uncertainty
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