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Advanced Air Mobility (AAM) is a growing field that demands accurate modeling of legal concepts and restrictions in navigating intelligent vehicles. In addition, any implementation of AAM needs to face the challenges posed by inherently dynamic and uncertain human-inhabited spaces robustly. Nevertheless, the employment of Unmanned Aircraft Systems (UAS) beyond visual line of sight (BVLOS) is an endearing task that promises to enhance significantly today's logistics and emergency response capabilities. To tackle these challenges, we present a probabilistic and neuro-symbolic architecture to encode legal frameworks and expert knowledge over uncertain spatial relations and noisy perception in an interpretable and adaptable fashion. More specifically, we demonstrate Probabilistic Mission Design (ProMis), a system architecture that links geospatial and sensory data with declarative, Hybrid Probabilistic Logic Programs (HPLP) to reason over the agent's state space and its legality. As a result, ProMis generates Probabilistic Mission Landscapes (PML), which quantify the agent's belief that a set of mission conditions is satisfied across its navigation space. Extending prior work on ProMis' reasoning capabilities and computational characteristics, we show its integration with potent machine learning models such as Large Language Models (LLM) and Transformer-based vision models. Hence, our experiments underpin the application of ProMis with multi-modal input data and how our method applies to many important AAM scenarios., Comment: arXiv admin note: text overlap with arXiv:2406.03454

Although Answer Set Programming (ASP) allows constraining neural-symbolic (NeSy) systems, its employment is hindered by the prohibitive costs of computing stable models and the CPU-bound nature of state-of-the-art solvers. To this end, we propose Answer Set Networks (ASN), a NeSy solver. Based on Graph Neural Networks (GNN), ASNs are a scalable approach to ASP-based Deep Probabilistic Logic Programming (DPPL). Specifically, we show how to translate ASPs into ASNs and demonstrate how ASNs can efficiently solve the encoded problem by leveraging GPU's batching and parallelization capabilities. Our experimental evaluations demonstrate that ASNs outperform state-of-the-art CPU-bound NeSy systems on multiple tasks. Simultaneously, we make the following two contributions based on the strengths of ASNs. Namely, we are the first to show the finetuning of Large Language Models (LLM) with DPPLs, employing ASNs to guide the training with logic. Further, we show the "constitutional navigation" of drones, i.e., encoding public aviation laws in an ASN for routing Unmanned Aerial Vehicles in uncertain environments., Comment: 16 pages, 9 figures

In a magnetic mirror fusion reactor, capturing the energy of fusion-produced alpha particles is essential to sustaining the reaction. However, since alpha particles are born at energies much higher than the confining potential, a substantial fraction are lost due to pitch-angle scattering before they can transfer their energy to the plasma via drag. The energy of prematurely lost alpha particles can still be captured through direct conversion, but designing an effective mechanism requires a description of the energies and times at which they become deconfined. Here we present the first-ever analytical solutions for the loss velocity, energy, and time distributions of alpha particles in a magnetic mirror. After obtaining the Fokker-Planck model of the Landau collision operator, we asymptotically solve for the orthogonal eigenfunctions of the Legendre operator with homogeneous boundary conditions to reveal an easily interpretable closed-form solution. Our framework is extremely general and can in principle be applied to any high-energy species, subject to any applied potential, in the low collisionality, modest potential regime for any mirror ratio R > 1, making this work broadly applicable to existing mirror devices., Comment: 12 pages, 15 figures

Collisions play a crucial role in governing particle and energy transport in plasmas confined in a magnetic mirror trap. Modern gyrokinetic codes are used to model transport in magnetic mirrors, but some of these codes utilize approximate model collision operators. This study focuses on a Pastukhov-style method of images calculation of particle and energy confinement times using a Lenard-Bernstein model collision operator. Prior work on parallel particle and energy balances used a different Fokker-Planck plasma collision operator and the method needs to be extended in non-trivial ways to study the Lenard-Bernstein operator. To assess the effectiveness of our approach, we compare our results with a modern finite element solver. Our findings reveal that the particle confinement time scales like $a \exp(a^2)$ using the Lenard-Bernstein operator, in contrast to the more accurate scaling that the Coulomb collision operator would yield $a^2 \exp(a^2)$, where $a^2$ is approximately proportional to the ambipolar potential. We propose that codes modeling collisional losses in a magnetic mirrors utilizing the Lenard-Bernstein or Dougherty collision operator scale their collision frequency of any electrostatically confined species. This study illuminates the intricate role the collision operator plays in the Pastukhov-style method of images calculation of collisional confinement., Comment: 21 pages, 4 figures, 3 tables, submitted to journal of plasma physics

Fault diagnosis is crucial for complex autonomous mobile systems, especially for modern-day autonomous driving (AD). Different actors, numerous use cases, and complex heterogeneous components motivate a fault diagnosis of the system and overall system integrity. AD systems are composed of many heterogeneous components, each with different functionality and possibly using a different algorithm (e.g., rule-based vs. AI components). In addition, these components are subject to the vehicle's driving state and are highly dependent. This paper, therefore, faces this problem by presenting the concept of a modular fault diagnosis framework for AD systems. The concept suggests modular state monitoring and diagnosis elements, together with a state- and dependency-aware aggregation method. Our proposed classification scheme allows for the categorization of the fault diagnosis modules. The concept is implemented on AD shuttle buses and evaluated to demonstrate its capabilities., Comment: Accepted at 2024 IEEE 20th International Conference on Intelligent Computer Communication and Processing (ICCP 2024)

How capable are diffusion models of generating synthetics texts? Recent research shows their strengths, with performance reaching that of auto-regressive LLMs. But are they also good in generating synthetic data if the training was under differential privacy? Here the evidence is missing, yet the promises from private image generation look strong. In this paper we address this open question by extensive experiments. At the same time, we critically assess (and reimplement) previous works on synthetic private text generation with LLMs and reveal some unmet assumptions that might have led to violating the differential privacy guarantees. Our results partly contradict previous non-private findings and show that fully open-source LLMs outperform diffusion models in the privacy regime. Our complete source codes, datasets, and experimental setup is publicly available to foster future research.

As cities strive to address urban mobility challenges, combining autonomous transportation technologies with intelligent infrastructure presents an opportunity to transform how people move within urban environments. Autonomous shuttles are particularly suited for adaptive and responsive public transport for the first and last mile, connecting with smart infrastructure to enhance urban transit. This paper presents the concept, implementation, and evaluation of a proof-of-concept deployment of an autonomous shuttle integrated with smart infrastructure at a public fair. The infrastructure includes two perception-equipped bus stops and a connected pedestrian intersection, all linked through a central communication and control hub. Our key contributions include the development of a comprehensive system architecture for "smart" bus stops, the integration of multiple urban locations into a cohesive smart transport ecosystem, and the creation of adaptive shuttle behavior for automated driving. Additionally, we publish an open source dataset and a Vehicle-to-X (V2X) driver to support further research. Finally, we offer an outlook on future research directions and potential expansions of the demonstrated technologies and concepts., Comment: Accepted by ECCV 2024 Workshop MAAS

Numerous Optimization Algorithms have a time-varying update rule thanks to, for instance, a changing step size, momentum parameter or, Hessian approximation. In this paper, we apply unrolled or automatic differentiation to a time-varying iterative process and provide convergence (rate) guarantees for the resulting derivative iterates. We adapt these convergence results and apply them to proximal gradient descent with variable step size and FISTA when solving partly smooth problems. We confirm our findings numerically by solving $\ell_1$ and $\ell_2$-regularized linear and logisitc regression respectively. Our theoretical and numerical results show that the convergence rate of the algorithm is reflected in its derivative iterates., Comment: arXiv admin note: text overlap with arXiv:2208.03107

In this paper, we propose two regularized proximal quasi-Newton methods with symmetric rank-1 update of the metric (SR1 quasi-Newton) to solve non-smooth convex additive composite problems. Both algorithms avoid using line search or other trust region strategies. For each of them, we prove a super-linear convergence rate that is independent of the initialization of the algorithm. The cubic regularized method achieves a rate of order $\left(\frac{C}{N^{1/2}}\right)^{N/2}$, where $N$ is the number of iterations and $C$ is some constant, and the other gradient regularized method shows a rate of the order $\left(\frac{C}{N^{1/4}}\right)^{N/2}$. To the best of our knowledge, these are the first global non-asymptotic super-linear convergence rates for regularized quasi-Newton methods and regularized proximal quasi-Newton methods. The theoretical properties are also demonstrated in two applications from machine learning.

Convergence in learning-to-optimize is hardly studied, because conventional convergence guarantees in optimization are based on geometric arguments, which cannot be applied easily to learned algorithms. Thus, we develop a probabilistic framework that resembles deterministic optimization and allows for transferring geometric arguments into learning-to-optimize. Our main theorem is a generalization result for parametric classes of potentially non-smooth, non-convex loss functions and establishes the convergence of learned optimization algorithms to stationary points with high probability. This can be seen as a statistical counterpart to the use of geometric safeguards to ensure convergence. To the best of our knowledge, we are the first to prove convergence of optimization algorithms in such a probabilistic framework.

Research on non-verbal behavior generation for social interactive agents focuses mainly on the believability and synchronization of non-verbal cues with speech. However, existing models, predominantly based on deep learning architectures, often perpetuate biases inherent in the training data. This raises ethical concerns, depending on the intended application of these agents. This paper addresses these issues by first examining the influence of gender on facial non-verbal behaviors. We concentrate on gaze, head movements, and facial expressions. We introduce a classifier capable of discerning the gender of a speaker from their non-verbal cues. This classifier achieves high accuracy on both real behavior data, extracted using state-of-the-art tools, and synthetic data, generated from a model developed in previous work.Building upon this work, we present a new model, FairGenderGen, which integrates a gender discriminator and a gradient reversal layer into our previous behavior generation model. This new model generates facial non-verbal behaviors from speech features, mitigating gender sensitivity in the generated behaviors. Our experiments demonstrate that the classifier, developed in the initial phase, is no longer effective in distinguishing the gender of the speaker from the generated non-verbal behaviors.

We present a probabilistic model for stochastic iterative algorithms with the use case of optimization algorithms in mind. Based on this model, we present PAC-Bayesian generalization bounds for functions that are defined on the trajectory of the learned algorithm, for example, the expected (non-asymptotic) convergence rate and the expected time to reach the stopping criterion. Thus, not only does this model allow for learning stochastic algorithms based on their empirical performance, it also yields results about their actual convergence rate and their actual convergence time. We stress that, since the model is valid in a more general setting than learning-to-optimize, it is of interest for other fields of application, too. Finally, we conduct five practically relevant experiments, showing the validity of our claims.

In this paper, we aim to study non-convex minimization problems via second-order (in-time) dynamics, including a non-vanishing viscous damping and a geometric Hessian-driven damping. Second-order systems that only rely on a viscous damping may suffer from oscillation problems towards the minima, while the inclusion of a Hessian-driven damping term is known to reduce this effect without explicit construction of the Hessian in practice. There are essentially two ways to introduce the Hessian-driven damping term: explicitly or implicitly. For each setting, we provide conditions on the damping coefficients to ensure convergence of the gradient towards zero. Moreover, if the objective function is definable, we show global convergence of the trajectory towards a critical point as well as convergence rates. Besides, in the autonomous case, if the objective function is Morse, we conclude that the trajectory converges to a local minimum of the objective for almost all initializations. We also study algorithmic schemes for both dynamics and prove all the previous properties in the discrete setting under proper choice of the step-size., Comment: 40 pages, 4 figures

Recent work has shown that synchrotron emission from relativistic plasmas leads the electron distribution to form an anisotropic ring in momentum space, which can be unstable to both kinetic and hydrodynamic instabilities. Fundamental to these works was the assumption that the plasma was optically thin, allowing all emitted radiation to escape. Here, we examine the behavior of these instabilities as the plasma becomes more optically thick. To do this, we extend a recently-developed Fokker-Planck operator for synchrotron emission and absorption in mildly relativistic plasmas to fully relativistic plasmas. For a given set of plasma parameters, photons emitted by higher-energy electrons tend to be higher frequency, and thus more easily escape the plasma. As a result, the ratio of the photon emission rate (radiative drag) to absorption rate (radiative diffusion) for a given electron is extremely energy-dependent. Given this behavior, we determine the critical parameters that control the opacity, and show how the plasma gradually transitions to become more isotropic and stable at higher opacity., Comment: 11 pages, 6 figures

Synchrotron radiation losses are a significant cause of concern for high-temperature aneutronic fusion reactions such as proton-Boron 11. The fact that radiation losses occur primarily in the high-energy tail, where the radiation itself has a substantial impact on the electron distribution, necessitates a self-consistent approach to modeling the diffusion and drag induced by synchrotron absorption and emission. Furthermore, an accurate model must account for the fact that the radiation emission spectrum is momentum-dependent, and the plasma opacity is frequency-dependent. Here, we present a simple Fokker-Planck operator, built on a newly-solved-for blackbody synchrotron diffusion operator, which captures all relevant features of the synchrotron radiation. Focusing on magnetic mirror fusion plasmas, we show that significant suppression of the electron distribution occurs for relativistic values of the perpendicular electron momentum, which therefore emit much less radiation than predicted under the assumption of a Maxwell-Juttner distribution., Comment: 14 pages, 6 figures

Our approach is part of the close link between continuous dissipative dynamical systems and optimization algorithms. We aim to solve convex minimization problems by means of stochastic inertial differential equations which are driven by the gradient of the objective function. This will provide a general mathematical framework for analyzing fast optimization algorithms with stochastic gradient input. Our study is a natural extension of our previous work devoted to the first-order in time stochastic steepest descent. Our goal is to develop these results further by considering second-order stochastic differential equations in time, incorporating a viscous time-dependent damping and a Hessian-driven damping. To develop this program, we rely on stochastic Lyapunov analysis. Assuming a square-integrability condition on the diffusion term times a function dependant on the viscous damping, and that the Hessian-driven damping is a positive constant, our first main result shows that almost surely, there is convergence of the values, and states fast convergence of the values in expectation. Besides, in the case where the Hessian-driven damping is zero, we conclude with the fast convergence of the values in expectation and in almost sure sense, we also managed to prove almost sure weak convergence of the trajectory. We provide a comprehensive complexity analysis by establishing several new pointwise and ergodic convergence rates in expectation for the convex and strongly convex case., Comment: 27 pages. arXiv admin note: text overlap with arXiv:2403.16775

Charged particles interacting with electromagnetic waves have a portion of their energy tied up in wave-driven oscillations. When these waves are localized to the exhaust of linear magnetic confinement systems this ponderomotive effect can be utilized to enhance particle confinement. The same effect can be derived for particles moving via an $\mathbf{E} \times \mathbf{B}$ drift into a region of a static perturbation to the electromagnetic fields which has a large wave vector component in the direction of the motion. In this work we use a simplified slab model to self-consistently solve for the electromagnetic fields within the fluid flowing plasma of a static flute-like ($k_\parallel = 0)$ perturbation, and evaluate the resulting ponderomotive potential. We find that two types of perturbations can exist within the flowing plasma, which are an O wave and an X wave in the frame moving with the fluid. In the case of tenuous plasma, these perturbations are magnetostatic or electrostatic multipole-analog perpendicular to the guiding magnetic field in the lab frame, respectfully. For denser plasmas, the O wave-like perturbation is screened at the electron skin depth scale, and the X wave-like perturbation is a combination of a similar perpendicular electric perturbation and parallel magnetic perturbation. The ponderomotive potential generated in the X wave-like case is gyrofrequency-dependent, and can be used as either potential barriers or potential wells, depending on the direction of the flow velocity., Comment: 14 pages, 10 figures

Towards designing learned optimization algorithms that are usable beyond their training setting, we identify key principles that classical algorithms obey, but have up to now, not been used for Learning to Optimize (L2O). Following these principles, we provide a general design pipeline, taking into account data, architecture and learning strategy, and thereby enabling a synergy between classical optimization and L2O, resulting in a philosophy of Learning Optimization Algorithms. As a consequence our learned algorithms perform well far beyond problems from the training distribution. We demonstrate the success of these novel principles by designing a new learning-enhanced BFGS algorithm and provide numerical experiments evidencing its adaptation to many settings at test time.

Quasi-Newton methods refer to a class of algorithms at the interface between first and second order methods. They aim to progress as substantially as second order methods per iteration, while maintaining the computational complexity of first order methods. The approximation of second order information by first order derivatives can be expressed as adopting a variable metric, which for (limited memory) quasi-Newton methods is of type ``identity $\pm$ low rank''. This paper continues the effort to make these powerful methods available for non-smooth systems occurring, for example, in large scale Machine Learning applications by exploiting this special structure. We develop a line search variant of a recently introduced quasi-Newton primal-dual algorithm, which adds significant flexibility, admits larger steps per iteration, and circumvents the complicated precalculation of a certain operator norm. We prove convergence, including convergence rates, for our proposed method and outperform related algorithms in a large scale image deblurring application.

Real world testing is of vital importance to the success of automated driving. While many players in the business design purpose build testing vehicles, we designed and build a modular platform that offers high flexibility for any kind of scenario. CoCar NextGen is equipped with next generation hardware that addresses all future use cases. Its extensive, redundant sensor setup allows to develop cross-domain data driven approaches that manage the transfer to other sensor setups. Together with the possibility of being deployed on public roads, this creates a unique research platform that supports the road to automated driving on SAE Level 5.

We use the PAC-Bayesian theory for the setting of learning-to-optimize. To the best of our knowledge, we present the first framework to learn optimization algorithms with provable generalization guarantees (PAC-Bayesian bounds) and explicit trade-off between convergence guarantees and convergence speed, which contrasts with the typical worst-case analysis. Our learned optimization algorithms provably outperform related ones derived from a (deterministic) worst-case analysis. The results rely on PAC-Bayesian bounds for general, possibly unbounded loss-functions based on exponential families. Then, we reformulate the learning procedure into a one-dimensional minimization problem and study the possibility to find a global minimum. Furthermore, we provide a concrete algorithmic realization of the framework and new methodologies for learning-to-optimize, and we conduct four practically relevant experiments to support our theory. With this, we showcase that the provided learning framework yields optimization algorithms that provably outperform the state-of-the-art by orders of magnitude.

Most automated driving functions are designed for a specific task or vehicle. Most often, the underlying architecture is fixed to specific algorithms to increase performance. Therefore, it is not possible to deploy new modules and algorithms easily. In this paper, we present our automated driving stack which combines both scalability and adaptability. Due to the modular design, our stack allows for a fast integration and testing of novel and state-of-the-art research approaches. Furthermore, it is flexible to be used for our different testing vehicles, including modified EasyMile EZ10 shuttles and different passenger cars. These vehicles differ in multiple ways, e.g. sensor setups, control systems, maximum speed, or steering angle limitations. Finally, our stack is deployed in real world environments, including passenger transport in urban areas. Our stack includes all components needed for operating an autonomous vehicle, including localization, perception, planning, controller, and additional safety modules. Our stack is developed, tested, and evaluated in real world traffic in multiple test sites, including the Test Area Autonomous Driving Baden-W\"urttemberg.

Motion planning is an essential part of autonomous mobile platforms. A good pipeline should be modular enough to handle different vehicles, environments, and perception modules. The planning process has to cope with all the different modalities and has to have a modular and flexible design. But most importantly, it has to be safe and robust. In this paper, we want to present our motion planning pipeline with particle swarm optimization (PSO) at its core. This solution is independent of the vehicle type and has a clear and simple-to-implement interface for perception modules. Moreover, the approach stands out for being easily adaptable to new scenarios. Parallel calculation allows for fast planning cycles. Following the principles of PSO, the trajectory planer first generates a swarm of initial trajectories that are optimized afterward. We present the underlying control space and inner workings. Finally, the application to real-world automated driving is shown in the evaluation with a deeper look at the modeling of the cost function. The approach is used in our automated shuttles that have already driven more than 3.500 km safely and entirely autonomously in sub-urban everyday traffic.

Our work is part of the close link between continuous-time dissipative dynamical systems and optimization algorithms, and more precisely here, in the stochastic setting. We aim to study stochastic convex minimization problems through the lens of stochastic inertial differential inclusions that are driven by the subgradient of a convex objective function. This will provide a general mathematical framework for analyzing the convergence properties of stochastic second-order inertial continuous-time dynamics involving vanishing viscous damping and measurable stochastic subgradient selections. Our chief goal in this paper is to develop a systematic and unified way that transfers the properties recently studied for first-order stochastic differential equations to second-order ones involving even subgradients in lieu of gradients. This program will rely on two tenets: time scaling and averaging, following an approach recently developed in the literature by one of the co-authors in the deterministic case. Under a mild integrability assumption involving the diffusion term and the viscous damping, our first main result shows that almost surely, there is weak convergence of the trajectory towards a minimizer of the objective function and fast convergence of the values and gradients. We also provide a comprehensive complexity analysis by establishing several new pointwise and ergodic convergence rates in expectation for the convex, strongly convex, and (local) Polyak-Lojasiewicz case. Finally, using Tikhonov regularization with a properly tuned vanishing parameter, we can obtain almost sure strong convergence of the trajectory towards the minimum norm solution., Comment: 31 pages. arXiv admin note: text overlap with arXiv:2403.06708

This paper focuses on language change based on shifting social norms, in particular with regard to the debate on language and gender. It is a recurring argument in this debate that language develops "naturally" and that "severe interventions" - such as gender-inclusive language is often claimed to be - in the allegedly "organic" language system are inappropriate and even "dangerous". Such interventions are, however, not unprecedented. Socially motivated processes of language change are neither unusual nor new. We focus in our contribution on one important political-social space in Germany, the German Bundestag. Taking other struggles about language and gender in the plenaries of the Bundestag as a starting point, our article illustrates that language and gender has been a recurring issue in the German Bundestag since the 1980s. We demonstrate how this is reflected in linguistic practices of the Bundestag, by the use of a) designations for gays and lesbians; b) pair forms such as B\"urgerinnen und B\"urger (female and male citizens); and c) female forms of addresses and personal nouns ('Pr\"asidentin' in addition to 'Pr\"asident'). Lastly, we will discuss implications of these earlier language battles for the currently very heated debate about gender-inclusive language, especially regarding new forms with gender symbols like the asterisk or the colon (Lehrer*innen, Lehrer:innen; male*female teachers) which are intended to encompass all gender identities., Comment: 40 pages, 9 figures

Research on gender and language is tightly knitted to social debates on gender equality and non-discriminatory language use. Psycholinguistic scholars have made significant contributions in this field. However, corpus-based studies that investigate these matters within the context of language use are still rare. In our study, we address the question of how much textual material would actually have to be changed if non-gender-inclusive texts were rewritten to be gender-inclusive. This quantitative measure is an important empirical insight, as a recurring argument against the use of gender-inclusive German is that it supposedly makes written texts too long and complicated. It is also argued that gender-inclusive language has negative effects on language learners. However, such effects are only likely if gender-inclusive texts are very different from those that are not gender-inclusive. In our corpus-linguistic study, we manually annotated German press texts to identify the parts that would have to be changed. Our results show that, on average, less than 1% of all tokens would be affected by gender-inclusive language. This small proportion calls into question whether gender-inclusive German presents a substantial barrier to understanding and learning the language, particularly when we take into account the potential complexities of interpreting masculine generics., Comment: 27 pages, 7 figures, 2 tables

In the limit of sufficiently fast rotation, rotating mirror traps are known to be stable against the loss-cone modes associated with conventional (non-rotating) mirrors. This paper calculates how quickly a mirror configuration must rotate in order for several of these modes to be stabilized (in particular, the high-frequency convective loss cone, drift cyclotron loss cone, and Dory-Guest-Harris modes). Commonalities in the stabilization conditions for these modes then motivate a modified formulation of the Gardner free energy and diffusively accessible free energy to be used for systems in which the important modes have wave vectors that are orthogonal or nearly orthogonal to the magnetic field, as well as a modification to include the effects of a loss region in phase space., Comment: 26 pages, 6 figures

The production of a heavy quark is accompanied by gluon bremsstrahlung with angular and momentum spectra predicted by perturbative Quantum Chromo Dynamics (QCD). The radiation off heavy quarks is predicted to be suppressed for large momentum particles, as a consequence of the angular ``dead cone effect''. In this paper, we studied this effect using data from Z boson decays to c- or b-quarks in $e^+e^-$ annihilation. The momentum spectra for charged particles are reconstructed in the momentum fraction variable $x$ or $\xi=\ln(1/x)$ by removing the decays of the heavy hadrons. We find an increasing suppression of particles with rising $x$ down to a fraction of $\lesssim 1/10$ for particles with $x\gtrsim0.2$ in b-quark and $x\gtrsim0.4$ in c-quark jets in comparison to light quark momentum spectra. The sensitivity to the dead cone effect in the present momentum analysis is larger than in the recently presented angular analysis. The suppression for c- and b-quark fragmentation is in good quantitative agreement with the expectations based on perturbative QCD within the Modified Leading Logarithmic Approximation (MLLA) in the central kinematic region. The data also support a two parameter description in the MLLA of these phenomena (``Limiting Spectrum''). The sensitivity of these measurements to the heavy quark mass is investigated., Comment: 6 pages, 6 figures. arXiv admin note: text overlap with arXiv:2303.13343

In order to minimize a differentiable geodesically convex function, we study a second-order dynamical system on Riemannian manifolds with an asymptotically vanishing damping term of the form $\alpha/t$. For positive values of $\alpha$, convergence rates for the objective values and convergence of trajectory is derived. We emphasize the crucial role of the curvature of the manifold for the distinction of the modes of convergence. There is a clear correspondence to the results that are known in the Euclidean case. When $\alpha$ is larger than a certain constant that depends on the curvature of the manifold, we improve the convergence rate of objective values compared to the previously known rate and prove the convergence of the trajectory of the dynamical system to an element of the set of minimizers. For $\alpha$ smaller than this curvature-dependent constant, the best known sub-optimal rates for the objective values and the trajectory are transferred to the Riemannian setting. We present computational experiments that corroborate our theoretical results.