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In 1939, Oppenheimer and Snyder showed that the continued gravitational collapse of a self-gravitating matter distribution can result in the formation of a black hole, cf.~ \cite{OS}. In this paper, which has greatly influenced the evolution of ideas around the concept of a black hole, matter was modeled as dust, a fluid with pressure equal to zero. We prove that when the corresponding initial data are suitably approximated by data for a collisionless gas as modeled by the Vlasov equation, then a trapped surface forms before the corresponding solution to the Einstein-Vlasov system can develop a singularity and again a black hole arises. As opposed to the dust case the pressure does not vanish for such solutions. As a necessary starting point for the analysis, which is carried out in Painlev\'{e}-Gullstrand coordinates, we prove a local existence and uniqueness theorem for regular solutions together with a corresponding extension criterion. The latter result will also become useful when one perturbs dust solutions containing naked singularities in the Vlasov framework., Comment: 74 pages

Experimentally measured characteristics of a kHz laser-driven Cu plasma X-ray source that was recently commissioned at ELI Beamlines facility are reported. The source can be driven either by an in-house developed high contrast sub-20 fs near-infrared TW laser based on optical parametric chirped-pulse amplification technology, or by a more conventional Ti:sapphire laser delivering 12 mJ, 45 fs pulses. The X-ray source parameters obtained with the two driving lasers are compared. Measured photon flux of the order up to 10^{12} K{\alpha} photons/4{\pi}/s is reported. Furthermore, experimental platforms for ultrafast X-ray diffraction and X-ray absorption and/or emission spectroscopy based on the reported source are described.

Let (X,D) be a projective log pair over the ring of integers of a number field such that the log canonical line bundle K_(X,D) or its dual -K_(X,D) is relatively ample. We introduce a canonical height of K_(X,D) (and -K(X,D)) which is finite precisely when the complexifications of K_(X,D) (and -K(X,D)) are K-semistable. When the complexifications are K-polystable, the canonical height is the height of K_(X,D) (and -K(X,D)) wrt any volume-normalized K\"ahler-Einstein metric on the complexifications of K_(X,D) (and -K(X,D)) The canonical height is shown to have a number of useful variational properties. Moreover, it may be expressed as a limit of periods on the N-fold products of the complexifications of X, as N tends to infinity. In particular, using this limit formula, the canonical height for the arithmetic log surfaces (P_1,D) over the integers, where D has at most three components, is computed explicitly in terms of the Hurwitz zeta function and its derivative at s=-1. Combining this explicit formula with previous height formulas for quaternionic Shimura curves yields a procedure for extracting information about the canonical integral models of some Shimura curves, such as wild ramification. Furthermore, explicit formulas for the canonical height of twisted Fermat curves are obtained, implying explicit Parshin type bounds for the Arakelov metric., Comment: 61 pages

The short-loading cycle is a repetitive task performed in high quantities, making it a great alternative for automation. In the short-loading cycle, an expert operator navigates towards a pile, fills the bucket with material, navigates to a dump truck, and dumps the material into the tipping body. The operator has to balance the productivity goal while minimising the fuel usage, to maximise the overall efficiency of the cycle. In addition, difficult interactions, such as the tyre-to-surface interaction further complicate the cycle. These types of hard-to-model interactions that can be difficult to address with rule-based systems, together with the efficiency requirements, motivate us to examine the potential of data-driven approaches. In this paper, the possibility of teaching an agent through reinforcement learning to approach a dump truck's tipping body and get in position to dump material in the tipping body is examined. The agent is trained in a 3D simulated environment to perform a simplified navigation task. The trained agent is directly transferred to a real vehicle, to perform the same task, with no additional training. The results indicate that the agent can successfully learn to navigate towards the dump truck with a limited amount of control signals in simulation and when transferred to a real vehicle, exhibits the correct behaviour.

Imaging radar is an emerging sensor modality in the context of Localization and Mapping (SLAM), especially suitable for vision-obstructed environments. This article investigates the use of 4D imaging radars for SLAM and analyzes the challenges in robust loop closure. Previous work indicates that 4D radars, together with inertial measurements, offer ample information for accurate odometry estimation. However, the low field of view, limited resolution, and sparse and noisy measurements render loop closure a significantly more challenging problem. Our work builds on the previous work - TBV SLAM - which was proposed for robust loop closure with 360$^\circ$ spinning radars. This article highlights and addresses challenges inherited from a directional 4D radar, such as sparsity, noise, and reduced field of view, and discusses why the common definition of a loop closure is unsuitable. By combining multiple quality measures for accurate loop closure detection adapted to 4D radar data, significant results in trajectory estimation are achieved; the absolute trajectory error is as low as 0.46 m over a distance of 1.8 km, with consistent operation over multiple environments., Comment: Submitted to the workshop "Radar in Robotics: Resilience from Signal to Navigation" at ICRA 2024

Motivated by conjectures in Mirror Symmetry, we continue the study of the real Monge--Amp\`ere operator on the boundary of a simplex. This can be formulated in terms of optimal transport, and we consider, more generally, the problem of optimal transport between symmetric probability measures on the boundary of a simplex and of the dual simplex. For suitably regular measures, we obtain regularity properties of the transport map, and of its convex potential. To do so, we exploit boundary regularity results for optimal transport maps by Caffarelli, together with the symmetries of the simplex., Comment: 13 pages

In 2001 Wolansky \cite{Wol} introduced a particle number-Casimir functional for the Einstein-Vlasov system. Two open questions are associated with this functional. First, a meaningful variational problem should be formulated and the existence of a minimizer to this problem should be established. The second issue is to show that a minimizer, for some choice of the parameters, is a static solution of the Einstein-Vlasov system. In the present work we solve the first problem by proving the existence of a minimizer to the particle number-Casimir functional. On the technical side, it is a main achievement that we are able to bypass the non-compactness of minimizing sequences by new arguments in both $v$-space and $x$-space, which might have several further applications. We note that such compactness results for the Einstein-Vlasov system have been absent in the literature, whereas similar results have been known in the Newtonian case. We also provide arguments which give strong support that minimizers corresponding to small masses are static solutions of the Einstein-Vlasov system. Furthermore, our analysis leads us to propose a new stability criterion for static solutions: We conjecture that a static solution for which the Casimir-binding energy is positive is stable for mass-preserving perturbations., Comment: 51 pages

In our previous work we conjectured - inspired by an algebro-geometric result of Fujita - that the height of an arithmetic Fano variety X of relative dimension $n$ is maximal when X is the projective space $\mathbb{P}^n_{\mathbb{Z}}$ over the integers, endowed with the Fubini-Study metric, if the corresponding complex Fano variety is K-semistable. In this work the conjecture is settled for diagonal hypersurfaces in $\mathbb{P}^{n+1}_{\mathbb{Z}}$. The proof is based on a logarithmic extension of our previous conjecture, of independent interest, which is established for toric log Fano varieties of relative dimension at most three, hyperplane arrangements on $\mathbb{P}^n_{\mathbb{Z}}$, as well as for general arithmetic orbifold Fano surfaces., Comment: 28 pages. Main changes in version 2: Included Lemma 4.3. In Thm 1.5 we now assume that the divisor has at most three irreducible components (which is automatic in the orbifold case). This assumption is needed in Lemma 6.2

The purpose of this work is to review the status about stationary solutions of the axially symmetric Einstein-Vlasov system with a focus on open problems of both analytical and numerical nature. For the latter we emphasize that the code used to construct stationary solutions in \cite{Ames2016,Ames2019} is open source, see \cite{Ames2023joss}. In the analytical setting the open problems include establishing methods for proving existence of axisymmetric stationary solutions which are far from spherically symmetric, both in the general case and for certain special classes of solutions pointed out in the text. In the numerical setting there are intriguing properties of highly relativistic solutions that demand further attention, such as whether a sequence of such stationary solutions can approach a Kerr black hole, or if they necessarily approach the thin ring limit reminiscent of cosmic strings. The question of whether stationary solutions include states with thin-disk like morphologies as seen in many galaxies is also open. Finally, there are opportunities to extend this research to new settings such as the case of massless particles and coupled black hole-matter systems. We believe that some of the open problems highlighted here are of central importance for the understanding of nature., Comment: 24 pages, 3 figures

We rigorously derive the quadrupole formula within the context of the Einstein-Vlasov system. The main contribution of this work is an estimate of the remainder terms, derived from well-defined assumptions, with explicitly stated error terms that depend on the solution's boundedness and decay properties, and the distance to the source. The assumptions are linked to established properties of global solutions of the Einstein-Vlasov system as in \cite{LT}. Prior derivations of the quadrupole formula have relied on post-Newtonian analysis and lacked comparisons with global solution properties. The importance of the no-incoming-radiation condition is emphasized underscoring the need for solutions satisfying this condition. This work thus addresses the limitations of existing results and provides motivation for further research on global solution properties of the Einstein-Vlasov system., Comment: 92 pages. To appear in Dissertationes Mathematicae

Factor graphs are a very powerful graphical representation, used to model many problems in robotics. They are widely spread in the areas of Simultaneous Localization and Mapping (SLAM), computer vision, and localization. In this paper we describe an approach to fill the gap with other areas, such as optimal control, by presenting an extension of Factor Graph Solvers to constrained optimization. The core idea of our method is to encapsulate the Augmented Lagrangian (AL) method in factors of the graph that can be integrated straightforwardly in existing factor graph solvers. We show the generality of our approach by addressing three applications, arising from different areas: pose estimation, rotation synchronization and Model Predictive Control (MPC) of a pseudo-omnidirectional platform. We implemented our approach using C++ and ROS. Besides the generality of the approach, application results show that we can favorably compare against domain specific approaches., Comment: This work has been submitted to the IEEE for possible publication

We consider gravitational collapse for the axially symmetric Einstein-Vlasov system. We investigate the weak cosmic censorship conjecture in the case of highly prolate initial data and we investigate the ``only if" part of the Hoop conjecture. Shapiro and Teukolsky initiated a similar study in 1991 \cite{Shapiro1991} where they found support that the weak cosmic censorship conjecture was violated for sufficiently prolate spheroidal initial data. More recently, independent studies of this problem have been carried out by Yoo, Harada and Okawa \cite{Yoo2017} and by East \cite{East2019}. A common feature in these works is that the initial data are dust-like. Dust can be considered as a \textit{singular} case of matter described by the Einstein-Vlasov system. The original motivation by Shapiro and Teukolsky to study this problem is based on the Lin-Mestel-Shu instability for gravitational collapse of uniform spheroids in the case of dust in Newtonian gravity. We argue that the Lin-Mestel-Shu solution is not relevant for studying the weak cosmic censorship of the Einstein-Vlasov system and we argue that dust-like initial data is also not relevant. To investigate collapse of highly prolate spheroidal configurations for the Einstein-Vlasov system is nevertheless interesting in view of the Hoop conjecture. By choosing highly prolate initial data the weak cosmic censorship conjecture is seriously tested. We carry out such a study for initial data which are not dust-like. We find formation of an apparent horizon in all cases we consider, which provides support for the weak cosmic censorship conjecture. In our tests of the Hoop conjecture we compute the polar circumference $\mathcal{C}_{H,p}$ at the time when the apparent horizon forms and find that it is less than 12% above $4\pi M$, where $M$ is the irreducible mass of the apparent horizon, which agrees with the spirit of the Hoop conjecture., Comment: 18 pages, 7 figures

We study the dynamics of avalanche ionization of pure helium nanodroplets activated by a weak extreme-ultraviolet (XUV) pulse and driven by an intense near-infrared (NIR) pulse. In addition to a transient enhancement of ignition of a nanoplasma at short delay times $\sim200$~fs, long-term activation of the nanodroplets lasting up to a few nanoseconds is observed. Molecular dynamics simulations suggest that the short-term activation is caused by the injection of seed electrons into the droplets by XUV photoemission. Long-term activation appears due to electrons remaining loosely bound to photoions which form stable `snowball' structures in the droplets. Thus, we show that XUV irradiation can induce long-lasting changes of the strong-field optical properties of nanoparticles, potentially opening new routes to controlling avalanche-ionization phenomena in nanostructures and condensed-phase systems.

Given a reflexive polytope with a height function, we prove a necessary and sufficient condition for solvability of the associated Monge-Amp\`ere equation. When the polytope is Delzant, solvability of this equation implies the metric SYZ conjecture for the corresponding family of Calabi-Yau hypersurfaces. We show how the location of the singularities in the tropical affine structure is determined by the PDE in the spirit of a free boundary problem and give positive and negative examples, demonstrating subtle issues with both solvability and properties of the singular set. We also improve on existing results regarding the SYZ conjecture for the Fermat family by showing regularity of the limiting potential., Comment: 46 pages. Added an appendix pointing out a geometric consequence of the regularity result for the standard unit simplex and the unit cube

Ultrafast electron dynamics drive phenomena such as photochemical reactions, catalysis, and light harvesting. To capture such dynamics in real-time, femtosecond to attosecond light sources are extensively used. However, an exact match between the excitation photon energy and a characteristic resonance is crucial. High-harmonic generation sources are exceptional in terms of pulse duration but limited in spectral tunability in the VUV range. Here, we present a monochromatic femtosecond source continuously tunable around 21 eV photon energy utilizing the second harmonic of an OPCPA laser system to drive high-harmonic generation. The unique tunability of the source is verified in an experiment probing the interatomic Coulombic decay in doped He nanodroplets across the He absorption bands. Moreover, we achieved intensities sufficient for driving non-linear processes using a tight focusing of the VUV beam. We demonstrated it on the observation of collective autoionization of multiply excited pure He nanodroplets., Comment: 12 pages, 5 figures, submitted to Nat. Commun

Robust SLAM in large-scale environments requires fault resilience and awareness at multiple stages, from sensing and odometry estimation to loop closure. In this work, we present TBV (Trust But Verify) Radar SLAM, a method for radar SLAM that introspectively verifies loop closure candidates. TBV Radar SLAM achieves a high correct-loop-retrieval rate by combining multiple place-recognition techniques: tightly coupled place similarity and odometry uncertainty search, creating loop descriptors from origin-shifted scans, and delaying loop selection until after verification. Robustness to false constraints is achieved by carefully verifying and selecting the most likely ones from multiple loop constraints. Importantly, the verification and selection are carried out after registration when additional sources of loop evidence can easily be computed. We integrate our loop retrieval and verification method with a fault-resilient odometry pipeline within a pose graph framework. By evaluating on public benchmarks we found that TBV Radar SLAM achieves 65% lower error than the previous state of the art. We also show that it's generalizing across environments without needing to change any parameters., Comment: Accepted for RAL, to be presented at IROS 2023, Detroit. Code: https://github.com/dan11003/tbv_slam_public Submission video: https://youtu.be/t8HQtHAUHHc

This paper presents an accurate, highly efficient, and learning-free method for large-scale odometry estimation using spinning radar, empirically found to generalize well across very diverse environments -- outdoors, from urban to woodland, and indoors in warehouses and mines - without changing parameters. Our method integrates motion compensation within a sweep with one-to-many scan registration that minimizes distances between nearby oriented surface points and mitigates outliers with a robust loss function. Extending our previous approach CFEAR, we present an in-depth investigation on a wider range of data sets, quantifying the importance of filtering, resolution, registration cost and loss functions, keyframe history, and motion compensation. We present a new solving strategy and configuration that overcomes previous issues with sparsity and bias, and improves our state-of-the-art by 38%, thus, surprisingly, outperforming radar SLAM and approaching lidar SLAM. The most accurate configuration achieves 1.09% error at 5Hz on the Oxford benchmark, and the fastest achieves 1.79% error at 160Hz., Comment: Published in Transactions on Robotics. Edited 2022-11-07: Updated affiliation and citation

A software architecture defines the blueprints of a large computational system, and is thus a crucial part of the design and development effort. This task has been explored extensively in the context of mobile robots, resulting in a plethora of reference designs and implementations. As the software architecture defines the framework in which all components are implemented, it is naturally a very important aspect of a mobile robot system. In this chapter, we overview the requirements that the particular problem domain (a mobile robot system) imposes on the software framework. We discuss some of the current design solutions, provide a historical perspective on common frameworks, and outline directions for future development., Comment: This chapter appears in: Ang, M.H., Khatib, O., Siciliano, B. (eds) Encyclopedia of Robotics. Springer, Berlin, Heidelberg

A sensor is a device that converts a physical parameter or an environmental characteristic (e.g., temperature, distance, speed, etc.) into a signal that can be digitally measured and processed to perform specific tasks. Mobile robots need sensors to measure properties of their environment, thus allowing for safe navigation, complex perception and corresponding actions, and effective interactions with other agents that populate it. Sensors used by mobile robots range from simple tactile sensors, such as bumpers, to complex vision-based sensors such as structured light RGB-D cameras. All of them provide a digital output (e.g., a string, a set of values, a matrix, etc.) that can be processed by the robot's computer. Such output is typically obtained by discretizing one or more analog electrical signals by using an Analog to Digital Converter (ADC) included in the sensor. In this chapter we present the most common sensors used in mobile robotics, providing an introduction to their taxonomy, basic features, and specifications. The description of the functionalities and the types of applications follows a bottom-up approach: the basic principles and components on which the sensors are based are presented before describing real-world sensors, which are generally based on multiple technologies and basic devices., Comment: This chapter appears in: Ang, M.H., Khatib, O., Siciliano, B. (eds) Encyclopedia of Robotics. Springer, Berlin, Heidelberg