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This paper deals with explicit upper and lower bounds for principal eigenvalues and the maximum principle associated to generalized Lane-Emden systems (GLE systems, for short). Regarding the bounds, we generalize the upper estimate of Berestycki, Nirenberg and Varadhan [Comm. Pure Appl. Math. (1994), 47-92] for the first eigenvalue of linear scalar problems on general domains to the case of strongly coupled GLE systems with $m \geqslant 2$ equations on smooth domains. The explicit lower estimate we obtain is also used to derive a maximum principle to GLE systems relying in terms of quantitative ingredients. Furthermore, as applications of the previous results, upper and lower estimates for the first eigenvalue of weighted poly-Laplacian eigenvalue problems with $L^p$ weights $(p>n)$ and Navier boundary condition are obtained. Moreover, a strong maximum principle depending on the domain and the weight function for scalar problems involving the poly-Laplacian operator is also established.

Graph generation is fundamental in diverse scientific applications, due to its ability to reveal the underlying distribution of complex data, and eventually generate new, realistic data points. Despite the success of diffusion models in this domain, those face limitations in sampling efficiency and flexibility, stemming from the tight coupling between the training and sampling stages. To address this, we propose DeFoG, a novel framework using discrete flow matching for graph generation. DeFoG employs a flow-based approach that features an efficient linear interpolation noising process and a flexible denoising process based on a continuous-time Markov chain formulation. We leverage an expressive graph transformer and ensure desirable node permutation properties to respect graph symmetry. Crucially, our framework enables a disentangled design of the training and sampling stages, enabling more effective and efficient optimization of model performance. We navigate this design space by introducing several algorithmic improvements that boost the model performance, consistently surpassing existing diffusion models. We also theoretically demonstrate that, for general discrete data, discrete flow models can faithfully replicate the ground truth distribution - a result that naturally extends to graph data and reinforces DeFoG's foundations. Extensive experiments show that DeFoG achieves state-of-the-art results on synthetic and molecular datasets, improving both training and sampling efficiency over diffusion models, and excels in conditional generation on a digital pathology dataset.

Binary asteroid formation is a highly complex process, which has been highlighted with recent observations of satellites with unexpected shapes, such as the oblate Dimorphos by the NASA DART mission and the contact binary Selam by NASA's Lucy mission. There is no clear consensus on which dynamical mechanisms determine the final shape of these objects. In turn, we explore a formation pathway where spin-up and rotational failure of a rubble pile asteroid lead to mass-shedding and a wide circumasteroidal debris disk in which the satellite forms. Using a combination of smooth-particle hydrodynamical and N-body simulations, we study the dynamical evolution in detail. We find that a debris disk containing matter corresponding to a few percent of the primary asteroid mass extending beyond the fluid Roche limit can consistently form both oblate and bilobate satellites via a series of tidal encounters with the primary body and mergers with other gravitational aggregates. Principally, satellites end up prolate (elongated) and on synchronous orbits, accreting mainly in a radial direction while tides from the primary asteroid keep the shape intact. However, close encounters and mergers can break the orbital state, leading to orbital migration and deformation. Satellite-satellite impacts occurring in this regime have lower impact velocities than merger-driven moon formation in e.g. planetary rings, leading to soft impacts between differently sized, non-spherical bodies. The resulting post-merger shape of the satellite is highly dependent on the impact geometry. Only moons having experienced a prior mild or catastrophic tidal disruption during a close encounter with the primary asteroid can become oblate spheroids, which is consistent with the predominantly prolate observed population of binary asteroid satellites., Comment: 37 pages, 24 figures, 3 tables. Accepted for publication in Icarus

Reactive graphs are transition structures whereas edges become active and inactive during its evolution, that were introduced by Dov Gabbay from a mathematical's perspective. This paper presents Marge (https://fm-dcc.github.io/MARGe), a web-based tool to visualise and analyse reactive graphs enriched with labels. Marge animates the operational semantics of reactive graphs and offers different graphical views to provide insights over concrete systems. We motivate the applicability of reactive graphs for adaptive systems and for featured transition systems, using Marge to tighten the gap between the existing theoretical models and their usage to analyse concrete systems., Comment: Companion paper of an article accepted at FACS 2024 with a similar name

The dynamics of non-equilibrium closed quantum systems and their route to thermalization are of fundamental interest to several fields, from cosmology to particle physics. However, a comprehensive description of non-equilibrium phenomena still presents a significant challenge. In this work, we report the observation of distinct stages during the relaxation of the decaying turbulence in trapped Bose-Einstein condensates. Our findings show a direct particle cascade from low to high momenta, a consequence of the energy injection in the system, exhibiting a characteristic universal scaling. This stage is followed by an inverse particle cascade responsible for repopulating the previously depleted condensate. Both cascades can be explained through self-similar solutions provided by wave turbulence theory. These findings provide important insights into the relaxation stages of out-of-equilibrium quantum many-body systems.

Strongly interacting systems appear in several areas of physics and are characterized by attractive interactions that can almost, or just barely, loosely bind two particles. Although this definition is made at the two-body level, this gives rise to fascinating effects in larger systems, including the so-called Efimov physics. In this context, the zero-range theory aims to describe low-energy properties based only on the scattering length. However, for a broad range of physical applications, the finite range of the interactions plays an important role. In this work, I discuss some aspects of finite-range effects in strongly interacting systems. I present the zero-range and shapeless universalities in two-body systems with applications in atomic and nuclear physics. I derived an analytical expression for the $s$-wave bound-state spectrum of the modified P\"oschl-Teller potential for two particles in three dimensions, which is compared with the approximations to illustrate their usefulness. Concerning three identical bosons, I presented a trimer energy scaling function that explicitly includes the effective range. The implications for larger systems are briefly discussed., Comment: 22 pages, 5 figures

Graph diffusion models have emerged as state-of-the-art techniques in graph generation, yet integrating domain knowledge into these models remains challenging. Domain knowledge is particularly important in real-world scenarios, where invalid generated graphs hinder deployment in practical applications. Unconstrained and conditioned graph generative models fail to guarantee such domain-specific structural properties. We present ConStruct, a novel framework that allows for hard-constraining graph diffusion models to incorporate specific properties, such as planarity or acyclicity. Our approach ensures that the sampled graphs remain within the domain of graphs that verify the specified property throughout the entire trajectory in both the forward and reverse processes. This is achieved by introducing a specific edge-absorbing noise model and a new projector operator. ConStruct demonstrates versatility across several structural and edge-deletion invariant constraints and achieves state-of-the-art performance for both synthetic benchmarks and attributed real-world datasets. For example, by leveraging planarity in digital pathology graph datasets, the proposed method outperforms existing baselines and enhances generated data validity by up to 71.1 percentage points.

Quantum Machine Learning investigates the possibility of quantum computers enhancing Machine Learning algorithms. Anomaly segmentation is a fundamental task in various domains to identify irregularities at sample level and can be addressed with both supervised and unsupervised methods. Autoencoders are commonly used in unsupervised tasks, where models are trained to reconstruct normal instances efficiently, allowing anomaly identification through high reconstruction errors. While quantum autoencoders have been proposed in the literature, their application to anomaly segmentation tasks remains unexplored. In this paper, we introduce a patch-based quantum autoencoder (QPB-AE) for image anomaly segmentation, with a number of parameters scaling logarithmically with patch size. QPB-AE reconstructs the quantum state of the embedded input patches, computing an anomaly map directly from measurement through a SWAP test without reconstructing the input image. We evaluate its performance across multiple datasets and parameter configurations and compare it against a classical counterpart.

With the advances in the optimization of magnetic field equilibria, stellarators have become a serious alternative to the tokamak, bringing this concept to the forefront of the pursuit of fusion energy. In order to be successful in experimentally demonstrating the viability of optimized stellarators, we must overcome any potential hurdles in the construction of its electromagnetic coils. Finding cost-effective ways of increasing the number of operating optimized stellarators could be key in cementing this magnetic confinement concept as a contender for a reactor. In this work, an alternative to modular coils, permanent magnets, are studied and are shown to enable the possibility of converting a tokamak into a stellarator. This is then applied to the case of ISTTOK tokamak, where an engineering design study is conducted., Comment: 34 pages, 24 figures

Target of NASA's DART mission, the system of Didymos and Dimorphos will once again be visited by a space mission -- ESA's Hera mission, scheduled to be launch in 2024. Hera will arrive in the system approximately 4 years after the DART impact, a long period compared to Dimorphos' orbital period (about 12 hours). It is therefore imperative to understand the dynamics of material in this environment on a long timescale. Here, we explore the long-term dynamics of the binary system (65038) Didymos, in the context of the perturbed, planar, circular and restricted 3-body problem. We design an analytical description for a symmetrical top-shaped object, the shape assumed for the Didymos, while the Dimorphos is considered an ellipsoid. In the absence of external effects, we identify seven stable equatorial regions where particles persist for more than a decade. However, in the presence of the solar radiation effect, the lifetime of small particles (

Dimorphos' oblate shape challenges formation models. Landslides on Didymos, induced by YORP effect, probably created a debris ring from which Dimorphos would have formed. Nonetheless, ring-derived satellites typically form with a prolate lemon shape. In light of the newest Dimorphos shape model, we revisit our previous work, Madeira et al. (2023a), and conducted new 1D simulations with material deposition in an extended region from 1.0 to 1.5 Didymos radii. An instantaneous landslide leads to a fast formation of a prolate Dimorphos directly from the ring. Now, if Didymos progressively deposits mass, Dimorphos grows through low-velocity impacts with large impactor-to-target mass ratio (pyramidal regime growth). Even during rapid, day-scale depositions, Dimorphos experiences one of these impacts, while for slower depositions, the satellite formation is primarily via pyramidal impacts. This process might reshape the satellite into an oblate shape (Leleu et al., 2018) or even in a contact-binary shape, a scenario worthy of investigation that should be studied in the future with more suitable tools. Our conclusions can be applied to Dinkinesh's satellite, recently discovered by NASA's Lucy mission., Comment: Note accepted for publication in Icarus journal

Graph-based representation approaches have been proven to be successful in the analysis of biomedical data, due to their capability of capturing intricate dependencies between biological entities, such as the spatial organization of different cell types in a tumor tissue. However, to further enhance our understanding of the underlying governing biological mechanisms, it is important to accurately capture the actual distributions of such complex data. Graph-based deep generative models are specifically tailored to accomplish that. In this work, we leverage state-of-the-art graph-based diffusion models to generate biologically meaningful cell-graphs. In particular, we show that the adopted graph diffusion model is able to accurately learn the distribution of cells in terms of their tertiary lymphoid structures (TLS) content, a well-established biomarker for evaluating the cancer progression in oncology research. Additionally, we further illustrate the utility of the learned generative models for data augmentation in a TLS classification task. To the best of our knowledge, this is the first work that leverages the power of graph diffusion models in generating meaningful biological cell structures.

We are concerned with the existence of solution of the problem $ -\Delta ^H_pu+|u|^{p-2}u=\lambda|u|^{q-2}u+ |u|^{p^*-2}u\quad \mbox{in}\quad\Omega,$ $u>0\quad \mbox{in}\quad\Omega,$ $a(\nabla u)\cdot \nu =0\quad \mbox{on}\quad\partial \Omega,$ where $\Delta ^H_pu=\mbox{div\,}(a(\nabla u))$, with $a(\xi)=H^{p-1}(\xi)\nabla H(\xi),\, \xi \in \mathbb{R}^N,$ $N\geqslant3,$ is the anisotropic $p$-Laplacian with $10$ is a parameter, and $p < q

Recent progress in out-of-equilibrium closed quantum systems has significantly advanced the understanding of mechanisms behind their evolution towards thermalization. Notably, the concept of non-thermal fixed points (NTFPs) - responsible for the emergence of spatio-temporal universal scaling in far-from-equilibrium systems - has played a crucial role in both theoretical and experimental investigations. In this work, we introduce a differential equation that has the universal scaling associated with NTFPs as a solution. The advantage of working with a differential equation, rather than only with its solution, is that we can extract several insightful properties not necessarily present in the solution alone. How the differential equation is derived allows physical interpretation of the universal exponents in terms of the time dependence of the amplitude of the distributions and their momentum scaling. Employing two limiting cases of the equation, we determined the universal exponents related to the scaling using the distributions near just two momentum values. We established a solid agreement with previous investigations by validating this approach with three distinct physical systems. This consistency highlights the universal nature of scaling due to NTFPs and emphasizes the predictive capabilities of the proposed differential equation. Moreover, under specific conditions, the equation predicts a power-law related to the ratio of the two universal exponents, leading to implications concerning particle and energy transport. This suggests that the observed power-laws in far-from-equilibrium turbulent fluids could be related to the universal scaling due to NTFPs, potentially offering new insights into the study of turbulence., Comment: 8 pages, 6 figures

Chariklo has two narrow and dense rings, C1R and C2R, located at 391 km and 405 km, respectively. In the light of new stellar occultation data, we study the stability around Chariklo. We also analyse three confinement mechanisms, to prevent the spreading of the rings, based on shepherd satellites in resonance with the edges of the rings. This study is made through a set of numerical simulations and the Poincar\'e surface of section technique. From the numerical simulation results we verify that, from the current parameters referring to the shape of Chariklo, the inner edge of the stable region is much closer to Chariklo than the rings. The Poincar\'e surface of sections allow us to identify the first kind periodic and quasi-periodic orbits, and also the resonant islands corresponding to the 1:2, 2:5, and 1:3 resonances. We construct a map of a versus e space which gives the location and width of the stable region and the 1:2, 2:5, and 1:3 resonances. We found that the first kind periodic orbits family can be responsible for a stable region whose location and size meet that of C1R, for specific values of the ring particles' eccentricities. However, C2R is located in an unstable region if the width of the ring is assumed to be about 120 m. After analysing different systems we propose that the best confinement mechanism is composed of three satellites, two of them shepherding the inner edge of C1R and the outer edge of C2R, while the third satellite would be trapped in the 1:3 resonance., Comment: Accepted for publication in Astronomy & Astrophysics