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This paper addresses a two-dimensional sharp interface variational model for solid-state dewetting of thin films with surface energies, introduced by Wang, Jiang, Bao, and Srolovitz in \cite{jiang2016solid}. Using the $H^{-1}$-gradient flow structure of the evolution law, short-time existence for a surface diffusion evolution equation with curvature regularization is established in the context of epitaxially strained two-dimensional films. The main novelty, as compared to the study of the wetting regime, is the presence of moving contact lines.

Single-molecule localization microscopy generates point clouds corresponding to fluorophore localizations. Spatial cluster identification and analysis of these point clouds are crucial for extracting insights about molecular organization. However, this task becomes challenging in the presence of localization noise, high point density, or complex biological structures. Here, we introduce MIRO (Multimodal Integration through Relational Optimization), an algorithm that uses recurrent graph neural networks to transform the point clouds in order to improve clustering efficiency when applying conventional clustering techniques. We show that MIRO supports simultaneous processing of clusters of different shapes and at multiple scales, demonstrating improved performance across varied datasets. Our comprehensive evaluation demonstrates MIRO's transformative potential for single-molecule localization applications, showcasing its capability to revolutionize cluster analysis and provide accurate, reliable details of molecular architecture. In addition, MIRO's robust clustering capabilities hold promise for applications in various fields such as neuroscience, for the analysis of neural connectivity patterns, and environmental science, for studying spatial distributions of ecological data.

We investigate the stability with respect to homogenization of classes of integrals arising in the control-theoretic interpretation of some Hamilton-Jacobi equations. The prototypical case is the homogenization of energies with a Lagrangian consisting of the sum of a kinetic term and a highly oscillatory potential $V =V_{\rm per}+ W$, where $V_{\rm per}$ is periodic and $W$ is a nonnegative perturbation thereof. We assume that $W$ has zero average in tubular domains oriented along a dense set of directions. Stability then holds true; that is, the resulting homogenized functional is identical to that for $W= 0$. We consider various extensions of this case. As a consequence of our results, we obtain stability for the homogenization of some steady-state and time-dependent, first-order Hamilton-Jacobi equations with convex Hamiltonians and perturbed periodic potentials. Finally, we show with an example that, for negative $W$, stability may not hold. Our study revisits and, depending on the different assumptions, complements results obtained by P.-L. Lions and collaborators using PDE techniques.

The observation of a resonance structure in the opening angle of the electron-positron pairs in the $^{7}$Li(p,\ee) $^{8}$Be reaction was claimed and interpreted as the production and subsequent decay of a hypothetical particle (X17). Similar excesses, consistent with this particle, were later observed in processes involving $^{4}$He and $^{12}$C nuclei with the same experimental technique. The MEG II apparatus at PSI, designed to search for the $\mu^+ \rightarrow \mathrm{e}^+ \gamma$ decay, can be exploited to investigate the existence of this particle and study its nature. Protons from a Cockroft-Walton accelerator, with an energy up to 1.1 MeV, were delivered on a dedicated Li-based target. The $\gamma$ and the e$^{+}$e$^{-}$ pair emerging from the $^8\mathrm{Be}^*$ transitions were studied with calorimeters and a spectrometer, featuring a broader angular acceptance than previous experiments. We present in this paper the analysis of a four-week data-taking in 2023 with a beam energy of 1080 keV, resulting in the excitation of two different resonances with Q-value \SI{17.6}{\mega\electronvolt} and \SI{18.1}{\mega\electronvolt}. No significant signal was found, and limits at \SI{90}{\percent} C.L. on the branching ratios (relative to the $\gamma$ emission) of the two resonances to X17 were set, $R_{17.6} < 1.8 \times 10^{-6} $ and $R_{18.1} < 1.2 \times 10^{-5} $., Comment: 11 pages, 9 figures, submitted to EPJC

The results on $\Gamma$-limits of sequences of free-discontinuity functionals with bounded cohesive surface terms are extended to the case of vector-valued functions. In this framework, we prove an integral representation result for the $\Gamma$-limit, which is then used to study deterministic and stochastic homogenisation problems for this type of functionals., Comment: 53 pages

Within the context of creating new Socially Assistive Robots, emotion recognition has become a key development factor, as it allows the robot to adapt to the user's emotional state in the wild. In this work, we focused on the analysis of two voice recording Spanish datasets: ELRA-S0329 and EmoMatchSpanishDB. Specifically, we centered our work in the paralanguage, e.~g. the vocal characteristics that go along with the message and clarifies the meaning. We proposed the use of the DeepSpectrum method, which consists of extracting a visual representation of the audio tracks and feeding them to a pretrained CNN model. For the classification task, DeepSpectrum is often paired with a Support Vector Classifier --DS-SVC--, or a Fully-Connected deep-learning classifier --DS-FC--. We compared the results of the DS-SVC and DS-FC architectures with the state-of-the-art (SOTA) for ELRA-S0329 and EmoMatchSpanishDB. Moreover, we proposed our own classifier based upon Attention Mechanisms, namely DS-AM. We trained all models against both datasets, and we found that our DS-AM model outperforms the SOTA models for the datasets and the SOTA DeepSpectrum architectures. Finally, we trained our DS-AM model in one dataset and tested it in the other, to simulate real-world conditions on how biased is the model to the dataset.

We present our contribution to the 7th ABAW challenge at ECCV 2024, by utilizing a Dual-Direction Attention Mixed Feature Network (DDAMFN) for multitask facial expression recognition, we achieve results far beyond the proposed baseline for the Multi-Task ABAW challenge. Our proposal uses the well-known DDAMFN architecture as base to effectively predict valence-arousal, emotion recognition, and facial action units. We demonstrate the architecture ability to handle these tasks simultaneously, providing insights into its architecture and the rationale behind its design. Additionally, we compare our results for a multitask solution with independent single-task performance., Comment: 11 pages

In this work we examine the stability of some classes of integrals, and in particular with respect to homogenization. The prototypical case is the homogenization of quadratic energies with periodic coefficients perturbed by a term vanishing at infinity, which has been recently examined in the framework of elliptic PDE. We use localization techniques and higher-integrability Meyers-type results to provide a closure theorem by $\Gamma$-convergence within a large class of integral functionals. From such result we derive stability theorems in homogenization which comprise the case of perturbations with zero average on the whole space. The results are also extended to the stochastic case, and specialized to the $G$-convergence of operators corresponding to quadratic forms. A corresponding analysis is also carried on for non-symmetric operators using the localization properties of $H$-convergence. Finally, we treat the case of perforated domains with Neumann boundary condition, and their stability.

The MEG II experiment, based at the Paul Scherrer Institut in Switzerland, reports the result of a search for the decay $\mu^+\to e^+\gamma$ from data taken in the first physics run in 2021. No excess of events over the expected background is observed, yielding an upper limit on the branching ratio of B($\mu^+\to e^+\gamma$) < $7.5 \times 10^{-13}$ (90% C.L.). The combination of this result and the limit obtained by MEG gives B($\mu^+\to e^+\gamma$) < $3.1 \times 10^{-13}$ (90% C.L.), which is the most stringent limit to date. A ten-fold larger sample of data is being collected during the years 2022-2023, and data-taking will continue in the coming years., Comment: 10 pages, 6 figures. To be published in EPJC

The MEG II experiment, located at the Paul Scherrer Institut (PSI) in Switzerland, is the successor to the MEG experiment, which completed data taking in 2013. MEG II started fully operational data taking in 2021, with the goal of improving the sensitivity of the mu+ -> e+ gamma decay down to 6e-14 almost an order of magnitude better than the current limit. In this paper, we describe the operation and performance of the experiment and give a new estimate of its sensitivity versus data acquisition time., Comment: 42 pages, 55 figures. Submitted to EPJC

We construct a general class of modified Ellis wormholes, where one asymptotic Minkowski region is replaced by a bounded 2-sphere core, characterized by asymptotic finite areal radius. We pursue an in-depth analysis of the resulting geometry, outlining that geodesic completeness is guaranteed also when the radial function asymptotically shrinks to zero. Then, we study the evolution of scalar perturbations, bringing out how these geometric configurations can in principle affect the time-domain profiles of quasinormal modes, pointing out the distinctive features with respect to other black holes or wormholes geometries.

We investigate the time evolution of spherically symmetric boson stars in Palatini $f(\mathcal{R})$ gravity through Numerical Relativity computations. Employing a novel approach that establishes a correspondence between modified gravity with scalar matter and General Relativity with modified scalar matter, we are able to use the techniques of Numerical Relativity to simulate these systems. Specifically, we focus on the quadratic theory $f(\mathcal{R})=\mathcal{R}+\xi\mathcal{R}^2$ and compare the obtained solutions with those in General Relativity, exploring both positive and negative values of the coupling parameter $\xi$. Our findings reveal that boson stars in Palatini $f(\mathcal{R})$ gravity exhibit both stable and unstable evolutions. The latter give rise to three distinct scenarios: migration towards a stable configuration, complete dispersion, and gravitational collapse leading to the formation of a baby universe structure.

In breast surgical planning, accurate registration of MR images across patient positions has the potential to improve the localisation of tumours during breast cancer treatment. While learning-based registration methods have recently become the state-of-the-art approach for most medical image registration tasks, these methods have yet to make inroads into breast image registration due to certain difficulties-the lack of rich texture information in breast MR images and the need for the deformations to be diffeomophic. In this work, we propose learning strategies for breast MR image registration that are amenable to diffeomorphic constraints, together with early experimental results from in-silico and in-vivo experiments. One key contribution of this work is a registration network which produces superior registration outcomes for breast images in addition to providing diffeomorphic guarantees.

Wheeler automata were introduced in 2017 as a tool to generalize existing indexing and compression techniques based on the Burrows-Wheeler transform. Intuitively, an automaton is said to be Wheeler if there exists a total order on its states reflecting the co-lexicographic order of the strings labeling the automaton's paths; this property makes it possible to represent the automaton's topology in a constant number of bits per transition, as well as efficiently solving pattern matching queries on its accepted regular language. After their introduction, Wheeler automata have been the subject of a prolific line of research, both from the algorithmic and language-theoretic points of view. A recurring issue faced in these studies is the lack of large datasets of Wheeler automata on which the developed algorithms and theories could be tested. One possible way to overcome this issue is to generate random Wheeler automata. Motivated by this observation, in this paper we initiate the theoretical study of random Wheeler automata, focusing on the deterministic case (Wheeler DFAs -- WDFAs). We start by extending the Erd\H{o}s-R\'enyi random graph model to WDFAs, and proceed by providing an algorithm generating uniform WDFAs according to this model. Our algorithm generates a uniform WDFA with $n$ states, $m$ transitions, and alphabet's cardinality $\sigma$ in $O(m)$ expected time ($O(m\log m)$ worst-case time w.h.p.) and constant working space for all alphabets of size $\sigma \le m/\ln m$. As a by-product, we also give formulas for the number of distinct WDFAs and obtain that $ n\sigma + (n - \sigma) \log \sigma$ bits are necessary and sufficient to encode a WDFA with $n$ states and alphabet of size $\sigma$, up to an additive $\Theta(n)$ term. We present an implementation of our algorithm and show that it is extremely fast in practice, with a throughput of over 8 million transitions per second., Comment: 17 pages, 3 figures

The integration of emotional intelligence in machines is an important step in advancing human-computer interaction. This demands the development of reliable end-to-end emotion recognition systems. However, the scarcity of public affective datasets presents a challenge. In this literature review, we emphasize the use of generative models to address this issue in neurophysiological signals, particularly Electroencephalogram (EEG) and Functional Near-Infrared Spectroscopy (fNIRS). We provide a comprehensive analysis of different generative models used in the field, examining their input formulation, deployment strategies, and methodologies for evaluating the quality of synthesized data. This review serves as a comprehensive overview, offering insights into the advantages, challenges, and promising future directions in the application of generative models in emotion recognition systems. Through this review, we aim to facilitate the progression of neurophysiological data augmentation, thereby supporting the development of more efficient and reliable emotion recognition systems.

We prove an integral-representation result for limits of non-local quadratic forms on $H^1_0(\Omega)$, with $\Omega$ a bounded open subset of $\mathbb R^d$, extending the representation on $C^\infty_c(\Omega)$ given by the Beurling-Deny formula in the theory of Dirichlet forms. We give a counterexample showing that a corresponding representation may not hold if we consider analogous functionals in $W^{1,p}_0(\Omega)$, with $p\neq 2$ and $1

We consider equilibrium models of spherical boson stars in Palatini $f(\mathcal{R})=\mathcal{R}+\xi \mathcal{R}^2$ gravity and study their collapse when perturbed. The Einstein-Klein-Gordon system is solved using a recently established correspondence in an Einstein frame representation. We find that, in that frame, the endpoint is a nonrotating black hole surrounded by a quasi-stationary cloud of scalar field. However, the dynamics in the $f(\mathcal{R})$ frame is dramatically different. The innermost region of the collapsing object exhibits the formation of a finite-size, exponentially-expanding $\textit{ baby universe}$ connected with the outer (parent) universe via a minimal area surface (a throat or umbilical cord). Our simulations indicate that this surface is at all times hidden inside a horizon, causally disconnecting the baby universe from observers above the horizon. The implications of our findings in other areas of gravitational physics are also discussed.

We investigate the absorption properties of reflection-asymmetric wormholes constructed via the thin-shell formalism in Palatini $f({\cal R})$ gravity. Such wormholes come from the matching of two Reissner-Nordstr\"om spacetimes at a time-like hypersurface (shell), which, according to the junction conditions in Palatini $f({\cal R})$ gravity, can have positive or negative energy density. Using numerical methods we investigate several configurations that satisfy the junction conditions, and analyze how the parameters of the system affect the absorption spectra. We confirm that the absorption cross section of wormholes at low frequencies significantly departs from that of black holes, and observe that in configurations made out of two naked singularities, the absorption spectra exhibit new features due to the effective light ring associated to the wormhole throat. The possibility of observing the presence of resonances at high frequencies is also discussed., Comment: 18 pages, 13 figures, revtex4-2 style

In order to characterize the mechanisms governing the diffusion of particles in biological scenarios, it is essential to accurately determine their diffusive properties. To do so, we propose a machine learning method to characterize diffusion processes with time-dependent properties at the experimental time resolution. Our approach operates at the single-trajectory level predicting the properties of interest, such as the diffusion coefficient or the anomalous diffusion exponent, at every time step of the trajectory. In this way, changes in the diffusive properties occurring along the trajectory emerge naturally in the prediction, and thus allow the characterization without any prior knowledge or assumption about the system. We first benchmark the method on synthetic trajectories simulated under several conditions. We show that our approach can successfully characterize both abrupt and continuous changes in the diffusion coefficient or the anomalous diffusion exponent. Finally, we leverage the method to analyze experiments of single-molecule diffusion of two membrane proteins in living cells: the pathogen-recognition receptor DC-SIGN and the integrin $\alpha5\beta1$. The analysis allows us to characterize physical parameters and diffusive states with unprecedented accuracy, shedding new light on the underlying mechanisms., Comment: 17 pages, 9 figures, 1 table. Code is found in https://github.com/BorjaRequena/step

Given a bounded open set $\Omega\subset \mathbb{R}^n$, we study sequences of quadratic functionals on the Sobolev space $H^1_0(\Omega)$, perturbed by sequences of bounded linear functionals. We prove that their $\Gamma$-limits, in the weak topology of $H^1_0(\Omega)$, can always be written as the sum of a quadratic functional, a linear functional, and a non-positive constant. The classical theory of $G$- and $H$-convergence completely characterises the quadratic and linear parts of the $\Gamma$-limit and shows that their coefficients do not depend on $\Omega$. The constant, which instead depends on $\Omega$ and will be denoted by $-\nu(\Omega)$, plays an important role in the study of the limit behaviour of the energies of the solutions. The main result of this paper is that, passing to a subsequence, we can prove that $\nu$ coincides with a non-negative Radon measure on a sufficiently large collection of bounded open sets $\Omega$. Moreover, we exhibit an example that shows that the previous result cannot be obtained for every bounded open set. The specific form of this example shows that the compactness theorem for the localisation method in $\Gamma$-convergence cannot be easily improved.

We prove a compactness result with respect to $\Gamma$-convergence for a class of integral functionals which are expressed as a sum of a local and a non-local term. The main feature is that, under our hypotheses, the local part of the $\Gamma$-limit depends on the interaction between the local and non-local terms of the converging subsequence. The result is applied to concentration and homogenization problems.

Anatomical structures such as blood vessels in contrast-enhanced CT (ceCT) images can be challenging to segment due to the variability in contrast medium diffusion. The combined use of ceCT and contrast-free (CT) CT images can improve the segmentation performances, but at the cost of a double radiation exposure. To limit the radiation dose, generative models could be used to synthesize one modality, instead of acquiring it. The CycleGAN approach has recently attracted particular attention because it alleviates the need for paired data that are difficult to obtain. Despite the great performances demonstrated in the literature, limitations still remain when dealing with 3D volumes generated slice by slice from unpaired datasets with different fields of view. We present an extension of CycleGAN to generate high fidelity images, with good structural consistency, in this context. We leverage anatomical constraints and automatic region of interest selection by adapting the Self-Supervised Body Regressor. These constraints enforce anatomical consistency and allow feeding anatomically-paired input images to the algorithm. Results show qualitative and quantitative improvements, compared to stateof-the-art methods, on the translation task between ceCT and CT images (and vice versa)., Comment: Accepted at BMVC 2022

Permissionless blockchains such as Bitcoin have long been criticized for their high computational and storage overhead. Unfortunately, while a number of proposals address the energy consumption of existing Proof-of-Work deployments, little attention has been given so far to remedy the storage overhead incurred by those blockchains. In fact, it seems widely acceptable that full nodes supporting the blockchains have to volunteer hundreds of GBs of their storage, to store and verify all transactions exchanged in the system. In this paper, we explore the solution space to effectively reduce the storage footprint of Proof-of-Work based blockchains. To do so, we analyze, by means of thorough empirical measurements, how existing full blockchain nodes utilize data from the shared ledger to validate incoming transactions/blocks. Based on this analysis, we show that it is possible for full nodes to locally reduce their storage footprint to approximately 15 GB, without any modification to the underlying protocol. We also discuss other client-side strategies to further reduce the storage footprint while incurring negligible computational overhead on the nodes.

Stochastic resetting can be naturally understood as a renewal process governing the evolution of an underlying stochastic process. In this work, we formally derive well-known results of diffusion with resets from a renewal theory perspective. Parallel to the concepts from renewal theory, we introduce the conditioned backward and forward times for stochastic processes with resetting to be the times since the last and until the next reset, given that the current state of the system is known. We focus on studying diffusion under Markovian and non-Markovian resetting. For these cases, we find the conditioned backward and forward time PDFs, comparing them with numerical simulations of the process. In particular, we find that for power-law reset time PDFs with asymptotic form $\varphi(t)\sim t^{-1-\alpha}$, significant changes in the properties of the conditioned backward and forward times happen at half-integer values of $\alpha$. This is due to the composition between the long-time scaling of diffusion $P(x,t)\sim 1/\sqrt{t}$ and the reset time PDF., Comment: Submitted to Physical Review E

We study some non-local functionals on the Sobolev space $W^{1,p}_0(\Omega)$ involving a double integral on $\Omega\times\Omega$ with respect to a measure $\mu$. We introduce a suitable notion of convergence of measures on product spaces which implies a stability property in the sense of $\Gamma$-convergence of the corresponding functionals.

We consider a walker moving in a one-dimensional interval with absorbing boundaries under the effect of Markovian resettings to the initial position. The walker's motion follows a random walk characterized by a general waiting time distribution between consecutive short jumps. We investigate the existence of an optimal reset rate, which minimizes the mean exit passage time, in terms of the statistical properties of the waiting time probability. Generalizing previous results restricted to Markovian random walks, we here find that, depending on the value of the relative standard deviation of the waiting time probability, resetting can be either (i) never beneficial, (ii) beneficial depending on the distance of the reset to the boundary, or (iii) always beneficial.

Denoising is of utmost importance for the visualization and processing of images featuring low signal-to-noise ratio. Total variation methods are among the most popular techniques to perform this task improving the signal-to-noise ratio while preserving coherent intensity discontinuities. In this work, a novel method, termed maximum likelihood data, is proposed, endowing the total variation formulation with the capability to deal with noise-specific models and pre-processing stages for a certain image of interest. To do this, the data fidelity term is modified by means of a maximum likelihood estimator between the original and the denoised image. To assess the improvements of the proposed method with respect to the total variation formulation, we study the denoising of high-frequency ultrasonic images on in-silico and in-vivo setups. The proposed method delivered a better contrast, preservation and localization of the structures while diminishing the intensity bias of the total variation formulation for the multiplicative noise. The enhancement of medical images through denoising helps to improve the outcome of subsequently applied image processing such as registration and segmentation procedures.

Deep learning (DL) models have provided state-of-the-art performance in various medical imaging benchmarking challenges, including the Brain Tumor Segmentation (BraTS) challenges. However, the task of focal pathology multi-compartment segmentation (e.g., tumor and lesion sub-regions) is particularly challenging, and potential errors hinder translating DL models into clinical workflows. Quantifying the reliability of DL model predictions in the form of uncertainties could enable clinical review of the most uncertain regions, thereby building trust and paving the way toward clinical translation. Several uncertainty estimation methods have recently been introduced for DL medical image segmentation tasks. Developing scores to evaluate and compare the performance of uncertainty measures will assist the end-user in making more informed decisions. In this study, we explore and evaluate a score developed during the BraTS 2019 and BraTS 2020 task on uncertainty quantification (QU-BraTS) and designed to assess and rank uncertainty estimates for brain tumor multi-compartment segmentation. This score (1) rewards uncertainty estimates that produce high confidence in correct assertions and those that assign low confidence levels at incorrect assertions, and (2) penalizes uncertainty measures that lead to a higher percentage of under-confident correct assertions. We further benchmark the segmentation uncertainties generated by 14 independent participating teams of QU-BraTS 2020, all of which also participated in the main BraTS segmentation task. Overall, our findings confirm the importance and complementary value that uncertainty estimates provide to segmentation algorithms, highlighting the need for uncertainty quantification in medical image analyses. Finally, in favor of transparency and reproducibility, our evaluation code is made publicly available at: https://github.com/RagMeh11/QU-BraTS., Comment: Accepted for publication at the Journal of Machine Learning for Biomedical Imaging (MELBA): https://www.melba-journal.org/papers/2022:026.html

In April 2021, scientists active in muon physics met to discuss and work out the physics case for the new High-Intensity Muon Beams (HIMB) project at PSI that could deliver of order $10^{10}$\,s$^{-1}$ surface muons to experiments. Ideas and concrete proposals were further substantiated over the following months and assembled in the present document. The high intensities will allow for completely new experiments with considerable discovery potential and unique sensitivities. The physics case is outstanding and extremely rich, ranging from fundamental particle physics via chemistry to condensed matter research and applications in energy research and elemental analysis. In all these fields, HIMB will ensure that the facilities S$\mu$S and CHRISP on PSI's High Intensity Proton Accelerator complex HIPA remain world-leading, despite the competition of muon facilities elsewhere., Comment: 116 pages, 42 figures

The MEG experiment took data at the Paul Scherrer Institute in the years 2009--2013 to test the violation of the lepton flavour conservation law, which originates from an accidental symmetry that the Standard Model of elementary particle physics has, and published the most stringent limit on the charged lepton flavour violating decay ${\mu}^+ \rightarrow {\rm e}^+ \gamma$: BR(${\mu}^+ \rightarrow {\rm e}^+ \gamma$) $<4.2 \times 10^{-13}$ at 90% confidence level. The MEG detector has been upgraded in order to reach a sensitivity of $6\times10^{-14}$. The basic principle of MEG II is to achieve the highest possible sensitivity using the full muon beam intensity at the Paul Scherrer Institute ($7\times10^{7}$ muons/s) with an upgraded detector. The main improvements are better rate capability of all sub-detectors and improved resolutions while keeping the same detector concept. In this paper, we present the current status of the preparation, integration and commissioning of the MEG II detector in the recent engineering runs., Comment: 12 pages, 4 figures. The version of acceptance for Symmetry

We study the behaviour of the solutions to a dynamic evolution problem for a viscoelastic model with long memory, when the rate of change of the data tends to zero. We prove that a suitably rescaled version of the solutions converges to the solution of the corresponding stationary problem.

Visco-energetic solutions have been recently advanced as a new solution concept for rate-independent systems, alternative to energetic solutions/quasistatic evolutions and balanced viscosity solutions. In the spirit of this novel concept, we revisit the analysis of the variational model proposed by Francfort and Marigo for the quasi-static crack growth in brittle materials, in the case of antiplane shear. In this context, visco-energetic solutions can be constructed by perturbing the time incremental scheme for quasistatic evolutions by means of a viscous correction inspired by the term introduced by Almgren, Taylor, and Wang in the study of mean curvature flows. With our main result we prove the existence of a visco-energetic solution with a given initial crack. We also show that, if the cracks have a finite number of tips evolving smoothly on a given time interval, visco-energetic solutions comply with Griffith's criterion.

We explore equilibrium solutions of spherically symmetric boson stars in the Palatini formulation of $f(\mathcal{R})$ gravity. We account for the modifications introduced in the gravitational sector by using a recently established correspondence between modified gravity with scalar matter and general relativity with modified scalar matter. We focus on the quadratic theory $f(\mathcal{R})=R+\xi R^2$ and compare its solutions with those found in general relativity, exploring both positive and negative values of the coupling parameter $\xi$. As matter source, a complex, massive scalar field with and without self-interaction terms is considered. Our results show that the existence curves of boson stars in Palatini $f(\mathcal{R})$ gravity are fairly similar to those found in general relativity. Major differences are observed for negative values of the coupling parameter which results in a repulsive gravitational component for high enough scalar field density distributions. Adding self-interactions makes the degeneracy between $f(\mathcal{R})$ and general relativity even more pronounced, leaving very little room for observational discrimination between the two theories., Comment: 16 pages, 10 figures, RevTex4-1