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The nilpotent graph of a group $G$ is the simple and undirected graph whose vertices are the elements of $G$ and two distinct vertices are adjacent if they generate a nilpotent subgroup of $G$. Here we discuss some topological properties of the nilpotent graph of a finite group $G$. Indeed, we characterize finite solvable groups whose closed neighborhoods are nilpotent subgroups. Moreover, we study the connectivity of the graph $\Gamma(G)$ obtained removing all universal vertices from the nilpotent graph of $G$. Some upper bounds to the diameter of $\Gamma(G)$ are provided when $G$ belongs to some classes of groups., Comment: Added a reference to the paper by Burness, Lucchini, and Nemmi

While stochastic resetting (or total resetting) is less young and more established concept in stochastic processes, partial stochastic resetting (PSR) is a relatively new field. PSR means that, at random moments in time, a stochastic process gets multiplied by a factor between 0 and 1, thus approaching but not reaching the resetting position. In this paper, we present new results on PSR highlighting the main similarities and discrepancies with total resetting. Specifically, we consider both symmetric $\alpha$-stable L\'evy processes (L\'evy flights) and Brownian motion with PSR in arbitrary d dimensions. We derive explicit expressions for the propagator and its stationary measure, and discuss in detail their asymptotic behavior. Interestingly, while approaching to stationarity, a dynamical phase transition occurs for the Brownian motion, but not for L\'evy flights. We also analyze the behavior of the process around the resetting position and find significant differences between PSR and total resetting.

Promptable segmentation foundation models have emerged as a transformative approach to addressing the diverse needs in medical images, but most existing models require expensive computing, posing a big barrier to their adoption in clinical practice. In this work, we organized the first international competition dedicated to promptable medical image segmentation, featuring a large-scale dataset spanning nine common imaging modalities from over 20 different institutions. The top teams developed lightweight segmentation foundation models and implemented an efficient inference pipeline that substantially reduced computational requirements while maintaining state-of-the-art segmentation accuracy. Moreover, the post-challenge phase advanced the algorithms through the design of performance booster and reproducibility tasks, resulting in improved algorithms and validated reproducibility of the winning solution. Furthermore, the best-performing algorithms have been incorporated into the open-source software with a user-friendly interface to facilitate clinical adoption. The data and code are publicly available to foster the further development of medical image segmentation foundation models and pave the way for impactful real-world applications., Comment: CVPR 2024 MedSAM on Laptop Competition Summary: https://www.codabench.org/competitions/1847/

There is a well established theory that links semi-Markov chains having Mittag-Leffler waiting times to time-fractional equations. We here go beyond the semi-Markov setting, by defining some non-Markovian chains whose waiting times, although marginally Mittag-Leffler, are assumed to be stochastically dependent. This creates a long memory tail in the evolution, unlike what happens for semi-Markov processes. As a special case of our chains, we study a particular counting process which extends the well-known fractional Poisson process, the last one having independent, Mittag-Leffler waiting times., Comment: 23 pages, 2 figures

Shaping the structure of light with flat optical devices has driven significant advancements in our fundamental understanding of light and light-matter interactions, and enabled a broad range of applications, from image processing and microscopy to optical communication, quantum information processing, and the manipulation of microparticles. Yet, pushing the boundaries of structured light beyond the linear optical regime remains an open challenge. Nonlinear optical interactions, such as wave mixing in nonlinear flat optics, offer a powerful platform to unlock new degrees of freedom and functionalities for generating and detecting structured light. In this study, we experimentally demonstrate the non-trivial structuring of third-harmonic light enabled by the addition of total angular momentum projection in a nonlinear, isotropic flat optics element -- a single thin film of amorphous silicon. We identify the total angular momentum projection and helicity as the most critical properties for analyzing the experimental results. The theoretical model we propose, supported by numerical simulations, offers quantitative predictions for light structuring through nonlinear wave mixing under various pumping conditions, including vectorial and non-paraxial pump light. Notably, we reveal that the shape of third-harmonic light is highly sensitive to the polarization state of the pump. Our findings demonstrate that harnessing the addition of total angular momentum projection in nonlinear wave mixing can be a powerful strategy for generating and detecting precisely controlled structured light.

Ultrafast excitation of non-equilibrium carriers under intense pulses offer unique opportunities for controlling optical properties of semiconductor materials. In this work, we propose a scheme for ultrafast generation of surface plasmon polaritons (SPPs) via a transient metagrating formed under two interfering optical pump pulses in the semiconductor GaAs thin film. The grating can be formed due to modulation of the refractive index associated with the non-equilibrium carriers generation. The formed temporal grating structure enables generation of SPP waves at GaAs/Ag interface via weak probe pulse excitation. We propose a theoretical model describing non-equilibrium carriers formation and diffusion and their contribution to permittivity modulation via Drude and band-filling mechanisms. We predict that by tuning the parameters of the pump and probe one can reach critical coupling regime and achieve efficient generation of SPP at the times scales of 0.1-1 ps.

Layered van der Waals (vdW) materials have emerged as a promising platform for nanophotonics due to large refractive indexes and giant optical anisotropy. Unlike conventional dielectrics and semiconductors, the absence of covalent bonds between layers allows for novel degrees of freedom in designing optically resonant nanophotonic structures down to the atomic scale, from the precise stacking of vertical heterostructures to controlling the twist angle between crystallographic axes. Specifically, while transition metal dichalcogenides monolayers exhibit giant second order nonlinear responses, their bulk counterparts with 2H stacking have zero second order response. In this work, we show second harmonic generation (SHG) arising from the interface of WS$_2$/MoS$_2$ hetero-bilayer thin films with an additional SHG enhancement in nanostructured optical antennas mediated by both the excitonic resonances and the anapole condition. When both conditions are met, we observe up to $10^2$ SHG signal enhancement. Our results highlights vdW materials as a platform for designing unique multilayer optical nanostructures and metamaterial, paving the way for advanced applications in nanophotonics and nonlinear optics., Comment: Manuscript + Supplementary (21 pages, 3 Main figures 8 supplementary figures)

Dry synthesis is a highly versatile method for the fabrication of nanoporous metal films, since it enables easy and reproducible deposition of single or multi-layer(s) of nanostructured materials that can find intriguing applications in plasmonics, photochemistry and photocatalysis, to name a few. Here, we extend the use of this methodology to the preparation of copper nanoislands that represent an affordable and versatile example of disordered plasmonic substrate. We perform detailed characterizations of the system using several techniques such as spectroscopic ellipsometry, cathodoluminescence, electron energy loss spectroscopy, ultrafast pump-probe spectroscopy and second-harmonic generation with the aim to investigate the optical properties of these systems in an unprecedented systematic way. Our study represents the starting point for future applications of this new disordered plasmonic system ranging from sensing to photochemistry and photocatalysis.

An astonishing feature of higher-order quantum theory is that it can accommodate indefinite causal order. In the simplest bipartite setting, there exist signaling correlations for which it is fundamentally impossible to ascribe a definite causal order for the parties' actions. Moreover, the assumptions required to arrive at such a statement (local quantum transformations and well-behaved probabilities) result in a nontrivial set of correlations, whose boundary is, to date, uncharacterized. In this work, we investigate indefinite causal order in boxworld theories. We construct a higher-order theory whose descriptor is the generalized bit (gbit) - the natural successor of the classical bit and quantum qubit. By fixing the local transformations in boxworld and asking about the global causal structure, we find that we trivially recover the full set of two-way signaling correlations. In light of this, we motivate and propose two physical principles in order to limit the set of achievable correlations: nonsignaling preservation and no signaling without system exchange. We find that a higher-order boxworld theory that respects these physical principles leads to (i) a nontrivial set of achievable correlations and (ii) a violation of some causal inequalities that is higher than what can be achieved in higher-order quantum theory. These results lead us to conjecture that the set of correlations of our higher-order boxworld theory is an outer approximation to the set of correlations produced by higher-order quantum theory., Comment: 15 + 27 pages, 9 figures

Third-generation gravitational-wave detectors will observe up to millions of merging binary black holes. With such a vast dataset, stacking events into population analyses will arguably be more important than analyzing single sources. We present the first application of population-level Fisher-matrix forecasts tailored to third-generation gravitational-wave interferometers. We implement the formalism first derived by Gair et al. and explore how future experiments such as Einstein Telescope and Cosmic Explorer will constrain the distributions of black-hole masses, spins, and redshift. Third-generation detectors will be transformative, improving constraints on the population hyperparameters by several orders of magnitude compared to current data. At the same time, we highlight that a single third-generation observatory and a network of detectors will deliver qualitatively similar performances. Obtaining precise measurements of some population features (e.g. peaks in the mass spectrum) will require only a few months of observations while others (e.g. the fraction of binaries with aligned spins) will instead require years if not decades. We argue population forecasts of this kind should be featured in white papers and feasibility studies aimed at developing the science case of future gravitational-wave interferometers., Comment: 16 pages, 6 figures, 3 tables

Recently, the field of 3D medical segmentation has been dominated by deep learning models employing Convolutional Neural Networks (CNNs) and Transformer-based architectures, each with their distinctive strengths and limitations. CNNs are constrained by a local receptive field, whereas transformers are hindered by their substantial memory requirements as well as they data hungriness, making them not ideal for processing 3D medical volumes at a fine-grained level. For these reasons, fully convolutional neural networks, as nnUNet, still dominate the scene when segmenting medical structures in 3D large medical volumes. Despite numerous advancements towards developing transformer variants with subquadratic time and memory complexity, these models still fall short in content-based reasoning. A recent breakthrough is Mamba, a Recurrent Neural Network (RNN) based on State Space Models (SSMs) outperforming Transformers in many long-context tasks (million-length sequences) on famous natural language processing and genomic benchmarks while keeping a linear complexity.

We propose a theoretical model for a colloidal heat engine driven by a feedback protocol that is able to fully convert the net heat absorbed by the hot bath into extracted work. The feedback protocol, inspired by gambling strategies, executes a sudden quench at zero work cost when the particle position satisfies a specific first-passage condition. As a result, the engine enhances both power and efficiency with respect to a standard Carnot cycle, surpassing Carnot's efficiency at maximum power. Using first-passage and martingale theory, we derive analytical expressions for the power and efficiency far beyond the quasistatic limit and provide scaling arguments for their dependency with the cycle duration. Numerical simulations are in perfect agreement with our theoretical findings, and illustrate the impact of the data acquisition rate on the engine's performance., Comment: 20 pages (including supplemental material), 8 figures (including supplemental figures)

Digital signal processing is the cornerstone of several modern-day technologies, yet in multiple applications it faces critical bottlenecks related to memory and speed constraints. Thanks to recent advances in metasurface design and fabrication, light-based analog computing has emerged as a viable option to partially replace or augment digital approaches. Several light-based analog computing functionalities have been demonstrated using patterned flat optical elements, with great opportunities for integration in compact nanophotonic systems. So far, however, the available operations have been restricted to the linear regime, limiting the impact of this technology to a compactification of Fourier optics systems. In this paper, we introduce nonlinear operations to the field of metasurface-based analog optical processing, demonstrating that nonlinear optical phenomena, combined with nonlocality in flat optics, can be leveraged to synthesize kernels beyond linear Fourier optics, paving the way to a broad range of new opportunities. As a practical demonstration, we report the experimental synthesis of a class of nonlinear operations that can be used to realize broadband, polarization-selective analog-domain edge detection.

Compact objects observed in gravitational-wave astronomy so far always come in pairs and never individually. Identifying the two components of a binary system is a delicate operation that is often taken for granted. The labeling procedure (i.e. which is object "1" and which is object "2") effectively acts as systematics, or equivalently an unspecified prior, in gravitational-wave data inference. The common approach is to label the objects solely by their masses, on a sample-by-sample basis; while intuitive, this leads to degeneracies when binaries have comparable masses. Instead, we argue that object identification should be tackled using the posterior distribution as a whole. We frame the problem in terms of constrained clustering -- a flavor of semi-supervised machine learning -- and find that unfolding the labeling systematics can significantly impact, and arguably improve, our interpretation of the data. In particular, the precision of black-hole spin measurements improves by up to 50%, spurious multimodalities and tails tend to disappear, posteriors become closer to Gaussian distributions, and the identification of the nature of the object (i.e. black hole vs. neutron star) is facilitated. We estimate that about 10% of the LIGO/Virgo posterior samples are affected by this relabeling, i.e. they might have been attributed to the wrong compact object in the observed binaries., Comment: 5 pages, 3 figures + appendices

Quantum Key Distribution (QKD) promises to revolutionize the field of security in communication, with applications ranging from state secrets to personal data, making it a key player in the ongoing battle against cyber threats. Reference-Frame-Independent (RFI) QKD aims to simplify QKD implementations by allowing to reduce the requirements of alignment on a shared reference frame. This is done by performing two mutually unbiased measurements on the control states. In this work, we present a novel fully passive receiver for time-bin encoded RFI-QKD. Conversion of time-bin to polarization is employed to perform the required quantum measurement in a fully passive manner. Furthermore, to overcome experimental errors, we retrieved a complete description of our measurement apparatus by employing a recently introduced Quantum Detector Self-Characterization technique, without performing tomographic studies on the detection stage. In fact, the security analysis carried out in this work uses experimentally retrieved Positive Operator Valued Measurements, which consider our receiver defects, substituting the ideal expected operators and thus increasing the overall level of secrecy. Lastly, we conducted a proof-of-principle experiment that validated the feasibility of our method and its applicability to QKD applications., Comment: 13 pages, 6 figures, 1 table

One of the main statistical features of near-neutral atmospheric boundary layer (ABL) turbulence is the positive vertical velocity skewness $Sk_w$ above the roughness sublayer or the buffer region in smooth-walls. The $Sk_w$ variations are receiving renewed interest in many climate-related parameterizations of the ABL given their significance to cloud formation and to testing sub-grid schemes for Large Eddy Simulations (LES). The vertical variations of $Sk_w$ are explored here using high Reynolds number wind tunnel and flume experiments collected above smooth, rough, and permeable-walls in the absence of buoyancy and Coriolis effects. These laboratory experiments form a necessary starting point to probe the canonical structure of $Sk_w$ as they deal with a key limiting case (i.e. near-neutral conditions) that has received much less attention compared to its convective counterpart in atmospheric turbulence studies. Diagnostic models based on cumulant expansions, realizability constraints, and the now-popular constant mass flux approach routinely employed in the convective boundary layer as well as prognostic models based on third-order budgets are used to explain variations in $Sk_w$ for the idealized laboratory conditions. The failure of flux-gradient relations to model $Sk_w$ from the gradients of the vertical velocity variance $\sigma_w^2$ are explained and corrections based on models of energy transport offered. Novel links between the diagnostic and prognostic models are also featured, especially for the inertial term in the third order budget of the vertical velocity fluctuation. The co-spectral properties of $w'/\sigma_w$ versus $w'^2/\sigma_w^2$ are also presented for the first time to assess the dominant scales governing $Sk_w$ in the inner and outer layers, where $w'$ is the fluctuating vertical velocity and $\sigma_w$ is the vertical velocity standard deviation.

The remnant black hole from a binary coalescence emits ringdown gravitational waves characterized by quasinormal modes, which depend solely on the remnant's mass and spin. In contrast, the ringdown amplitudes and phases are determined by the properties of the merging progenitors. Accurately modeling these amplitudes and phases reduces systematic biases in parameter estimation and enables the development and performance of rigorous tests of general relativity. We present a state-of-the-art, data-driven surrogate model for ringdown amplitudes and phases, leveraging Gaussian process regression trained against SXS numerical-relativity simulations. Focusing on nonprecessing, quasicircular binary black holes, our model offers the most comprehensive fit that includes 16 emission modes, incorporating overtones and quadratic contributions. Our surrogate model achieves reconstruction errors that are approximately 2 orders of magnitude smaller than the typical measurement errors of current gravitational-wave interferometers. An additional benefit of our approach is its flexibility, which allows for future extensions to include features such as eccentricity and precession, broadening the scope of its applicability to more generic astrophysical scenarios. Finally, we are releasing our model in a ready-to-use package called postmerger., Comment: 12 pages, 7 figures; code usage examples are available at https://github.com/cpacilio/postmerger . v2: matches published version

The recent availability of quantum annealers as cloud-based services has enabled new ways to handle machine learning problems, and several relevant algorithms have been adapted to run on these devices. In a recent work, linear regression was formulated as a quadratic binary optimization problem that can be solved via quantum annealing. Although this approach promises a computational time advantage for large datasets, the quality of the solution is limited by the necessary use of a precision vector, used to approximate the real-numbered regression coefficients in the quantum formulation. In this work, we focus on the practical challenge of improving the precision vector encoding: instead of setting an array of generic values equal for all coefficients, we allow each one to be expressed by its specific precision, which is tuned with a simple adaptive algorithm. This approach is evaluated on synthetic datasets of increasing size, and linear regression is solved using the D-Wave Advantage quantum annealer, as well as classical solvers. To the best of our knowledge, this is the largest dataset ever evaluated for linear regression on a quantum annealer. The results show that our formulation is able to deliver improved solution quality in all instances, and could better exploit the potential of current quantum devices.

Input variables in numerical models are often subject to several levels of uncertainty, usually modeled by probability distributions. In the context of uncertainty quantification applied to these models, studying the robustness of output quantities with respect to the input distributions requires: (a) defining variational classes for these distributions; (b) calculating boundary values for the output quantities of interest with respect to these variational classes. The latter should be defined in a way that is consistent with the information structure defined by the ``baseline'' choice of input distributions. Considering parametric families, the variational classes are defined using the geodesic distance in their Riemannian manifold, a generic approach to such problems. Theoretical results and application tools are provided to justify and facilitate the implementation of such robustness studies, in concrete situations where these distributions are truncated -- a setting frequently encountered in applications. The feasibility of our approach is illustrated in a simplified industrial case study.

We construct three dimensional non-semisimple topological field theories from the unrolled quantum group of the Lie superalgebra $\mathfrak{osp}(1 \vert 2)$. More precisely, the quantum group depends on a root of unity $q=e^{\frac{2 \pi \sqrt{-1}}{r}}$, where $r$ is a positive integer greater than $2$, and the construction applies when $r$ is not congruent to $4$ modulo $8$. The algebraic result which underlies the construction is the existence of a relative modular structure on the non-finite, non-semisimple category of weight modules for the quantum group. We prove a Verlinde formula which allows for the computation of dimensions and Euler characteristics of topological field theory state spaces of unmarked surfaces. When $r$ is congruent to $\pm 1$ or $\pm 2$ modulo $8$, we relate the resulting $3$-manifold invariants with physicists' $\widehat{Z}$-invariants associated to $\mathfrak{osp}(1 \vert 2)$. Finally, we establish a relation between $\widehat{Z}$-invariants associated to $\mathfrak{sl}(2)$ and $\mathfrak{osp}(1 \vert 2)$ which was conjectured in the physics literature., Comment: 38 pages

We provide empirical evidence that within the inertial sub layer of adiabatic turbulent flows over smooth walls, the skewness of the vertical velocity component $Sk_w$ displays universal behaviour, being constant and constrained within the range $Sk_w \approx 0.1-0.16$, regardless of flow configuration and Reynolds number. A theoretical model is proposed to explain the observed behaviour, including the observed range of variations of $Sk_w$. The model clarifies why $Sk_w$ cannot be predicted from down-gradient closure approximations routinely employed in meteorological and climate models whereby $Sk_w$ impacts cloud formation and dispersion processes. The model also offers an alternative and implementable approach.

This paper presents the Nordic-Estonian Quantum Computing e-Infrastructure Quest - NordIQuEst - an international collaboration of scientific and academic organizations from Denmark, Estonia, Finland, Norway, and Sweden, working together to develop a hybrid High-Performance and Quantum Computing (HPC+QC) infrastructure. The project leverages existing and upcoming classical high-performance computing and quantum computing systems, facilitating the development of interconnected systems. Our effort pioneers a forward-looking architecture for both hardware and software capabilities, representing an early-stage development in hybrid computing infrastructure. Here, we detail the outline of the initiative, summarizing the progress since the project outset, and describing the framework established. Moreover, we identify the crucial challenges encountered, and potential strategies employed to address them., Comment: 15 pages, 2 figures

Real 3-manifold triangulations can be uniquely represented by isomorphism signatures. Databases of these isomorphism signatures are generated for a variety of 3-manifolds and knot complements, using SnapPy and Regina, then these language-like inputs are used to train various machine learning architectures to differentiate the manifolds, as well as their Dehn surgeries, via their triangulations. Gradient saliency analysis then extracts key parts of this language-like encoding scheme from the trained models. The isomorphism signature databases are taken from the 3-manifolds' Pachner graphs, which are also generated in bulk for some selected manifolds of focus and for the subset of the SnapPy orientable cusped census with $<8$ initial tetrahedra. These Pachner graphs are further analysed through the lens of network science to identify new structure in the triangulation representation; in particular for the hyperbolic case, a relation between the length of the shortest geodesic (systole) and the size of the Pachner graph's ball is observed., Comment: 35 pages; 23 figures; 7 tables

Geospatial data has been transformative for the monitoring of the Earth, yet, as in the case of (geo)physical monitoring, the measurements can have variable spatial and temporal sampling and may be associated with a significant level of perturbations degrading the signal quality. Denoising geospatial data is, therefore, essential, yet often challenging because the observations may comprise noise coming from different origins, including both environmental signals and instrumental artifacts, which are spatially and temporally correlated, thus hard to disentangle. This study addresses the denoising of multivariate time series acquired by irregularly distributed networks of sensors, requiring specific methods to handle the spatiotemporal correlation of the noise and the signal of interest. Specifically, our method focuses on the denoising of geodetic position time series, used to monitor ground displacement worldwide with centimeter- to-millimeter precision. Among the signals affecting GNSS data, slow slip events (SSEs) are of interest to seismologists. These are transients of deformation that are weakly emerging compared to other signals. Here, we design SSEdenoiser, a multi-station spatiotemporal graph-based attentive denoiser that learns latent characteristics of GNSS noise to reveal SSE-related displacement with sub-millimeter precision. It is based on the key combination of graph recurrent networks and spatiotemporal Transformers. The proposed method is applied to the Cascadia subduction zone, where SSEs occur along with bursts of tectonic tremors, a seismic rumbling identified from independent seismic recordings. The extracted events match the spatiotemporal evolution of tremors. This good space-time correlation of the denoised GNSS signals with the tremors validates the proposed denoising procedure.

We investigate entanglement detection in quantum materials through criteria based on the simultaneous suppression of collective matter excitations. Unlike other detection schemes, these criteria can be applied to continuous and unbounded variables. By considering a system of interacting dipoles on a lattice, we show the detection of collective entanglement arising from two different physical mechanisms, namely, the ferroelectric ordering and the dressing of matter degrees of freedom by light. In the latter case, the detection shows the formation of a collective entangled phase not directly related to spontaneous symmetry breaking. These results open a new perspective for the entanglement characterization of competing orders in quantum materials, and have direct application to quantum paraelectrics with large polariton splittings., Comment: 5 pages, 3 figures

Gravitational waves emitted by a ringing black hole allow us to perform precision tests of general relativity in the strong field regime. With improvements to our current gravitational wave detectors and upcoming next-generation detectors, developing likelihood-free parameter inference infrastructure is critical as we will face complications like nonstandard noise properties, partial data and incomplete signal modeling that may not allow for an analytically tractable likelihood function. In this work, we present a proof-of-concept strategy to perform likelihood-free Bayesian inference on ringdown gravitational waves using simulation based inference. Specifically, our method is based on truncated sequential neural posterior estimation, which trains a neural density estimator of the posterior for a specific observed data segment. We setup the ringdown parameter estimation directly in the time domain. We show that the parameter estimation results obtained using our trained networks are in agreement with well-established Markov-chain methods for simulated injections as well as analysis on real detector data corresponding to GW150914. Additionally, to assess our approach's internal consistency, we show that the density estimators pass a Bayesian coverage test., Comment: 12 pages, 7 figures; v2: minor changes, matches published version; v3: metadata added

Turbulent flows over a large surface area (S) covered by n obstacles experience an overall drag due to the presence of the ground and the protruding obstacles into the flow. The drag partition between the roughness obstacles and the ground is analyzed using an analytical model proposed by Raupach (1992) and is hereafter referred to as R92. The R92 is based on the premise that the wake behind an isolated roughness element can be described by a shelter area A and a shelter volume V. The individual sizes of A and V without any interference from other obstacles can be determined from scaling analysis for the spread of wakes. To upscale from an individual roughness element to n/S elements where wakes may interact, R92 adopted a background stress re-normalizing instead of reducing A or V with each element addition. This work shows that R92's approach only converges to a linear reduction in A and V for small n/S where wakes have low probability of interacting with one another. This probabilistic nature suggests that up-scaling from individual to multiple roughness elements can be re-formulated using stochastic averaging methods proposed here. The two approaches are shown to recover R92 under plausible conditions. Comparisons between R92 and available data on blocks and vegetation-like roughness elements confirm the practical utility of R92 and its potential use in large-scale models provided the relevant parameters accommodate certain features of the roughness element type (cube versus vegetation-like) and, to a lesser extent, their configuration throughout S.

The study of verbal subgroups within a group is well-known for being an effective tool to obtain structural information about a group. Therefore, conditions that allow the classification of words in a free group are of paramount importance. One of the most studied problems is to establish which words are concise, where a word $w$ is said to be concise if the verbal subgroup $w(G)$ is finite in each group $G$ in which $w$ takes only a finite number of values. The purpose of this article is to present some results, in which a hierarchy among words is introduced, generalizing the concept of concise word.