Abstract In this work, we explore spin addition techniques within the context of solid-state physics, focusing on systems with
magnetic impurities. A notable example of such systems is the emergence of the Kondo effect, which arises from the
coupling between conduction electrons and a single localized magnetic atom embedded in a metallic host. Computational approaches—such as the numerical renormalization group (NRG) and related renormalization techniques—are
frequently employed to study these models. These methods often rely on constructing an expanding basis of quantum states by iteratively adding spin degrees of freedom to a known set of states. To illustrate this procedure, we
present a pedagogical discussion on the step-by-step addition of spin-½ particles by means of a toy model, using Clebsch–Gordan coefficients. This approach not only serves a pedagogical purpose but also underscores the relevance of
Clebsch–Gordan coefficients in the construction of an expanding basis for computational renormalization. While we
do not aim to present novel physical results, our objective is to offer a didactic perspective on the iterative structure
underlying these computational techniques.

Owing to its substantial implications for black hole spectroscopy, spectral instability has attracted considerable attention in the literature. While the emergence of such instability is attributed to the non-Hermitian nature of the gravitational system, it remains sensitive to various factors. In this work, we conduct a focused analysis of black hole spectral instability using the Pöschl–Teller potential as a toy model. We investigate the dependence of the resulting spectral instability on the magnitude, spatial scale, and localization of deterministic and random perturbations in the effective potential of the wave equation, and discuss the underlying physical interpretations. It is observed that small perturbations in the potential initially have a limited impact on the less damped black hole quasinormal modes, with deviations typically around their unperturbed values, a phenomenon first derived by Skakala and Visser in a more restrictive context. In the higher-overtone region, the deviation propagates, amplifies, and eventually gives rise to spectral instability and, inclusively, bifurcation in the quasinormal mode spectrum. While deterministic perturbations give rise to a deformed but well-defined quasinormal spectrum, random perturbations lead to uncertainties in the resulting spectrum. Nonetheless, the primary trend of the spectral instability remains consistent, being sensitive to both the strength and location of the perturbation. However, we demonstrate that the observed spectral instability might be suppressed for perturbations that are physically appropriate.

In this paper we consider the extent to which a lack of observations from SETI may be used to quantify the Fermi paradox. We compute the probability of at least one detection of an extraterrestrial electromagnetic (EM) signal of Galactic origin, as a function of the number N of communicative civilizations. We show how this is derivable from the probability of detecting a single signal; the latter is ≈ 0.6 δ / R , where δ is the distance between the initial and final EM signals and R is the radius of the Milky Way, for δ / R ≪ 1 . We show how to combine this analysis with the Drake equation N = 𝒩 δ / c , where c is the speed of light; this implies, applying a simplified toy model as an example, that the probability of detecting at least one signal is > 99 % for δ / c ≳ 10 2.8 years, given that 𝒩 = 1 . Lastly, we list this toy model’s significant limitations, and suggest ways to ameliorate them in more realistic future models.

Neutrino-nucleus scattering cross sections are critical theoretical inputs for long-baseline neutrino oscillation experiments. However, robust modeling of these cross sections remains challenging. For a simple but physically motivated toy model of the DUNE experiment, we demonstrate that an accurate neural-network model of the cross section—leveraging only Standard-Model symmetries—can be learned from near-detector data. We perform a neutrino oscillation analysis with simulated far-detector events, finding that oscillation analysis results enabled by our data-driven cross-section model approach the theoretical limit achievable with perfect prior knowledge of the cross section. We further quantify the effects of flux shape and detector resolution uncertainties as well as systematics from cross-section mismodeling. This proof-of-principle study highlights the potential of future neutrino near-detector datasets and data-driven cross-section models.

Axion-like particles (ALPs) are often considered good candidates for dark matter (DM). Several mechanisms for generating the relic abundance of ALP DM have been proposed, involving processes that may occur either before, during, or after cosmic inflation. In all cases, the potential of the corresponding Peccei–Quinn (PQ) field plays an essential role. We investigate the radiative, thermal, and space-time curvature corrections to the PQ field dynamics in scenarios where the potential exhibits very small self-coupling. We focus on toy models with a quasi-supersymmetric spectrum and discuss how accounting for these corrections is crucial for obtaining reliable predictions for the relic abundance of ALP DM. Abstract Published by the Jagiellonian University 2025 authors

Numerical relativity simulations provide a means by which to study the evolution and end point of strong overdensities in cosmological spacetimes. Specific applications include studies of primordial black hole formation and the robustness of inflation. Here we adopt a toy model previously used in asymptotically flat spacetimes to show that, for given values of the overdensity and the mean curvature, solutions to the Hamiltonian constraint need not exist, and if they do exist they are not unique. Specifically, pairs of solutions exist on two branches, corresponding to strong-field and weak-field solutions, that join at a maximum beyond which solutions cease to exist. As a result, there is a limit to the extent to which an overdensity can be balanced by intrinsic rather than extrinsic curvature on the initial slice. Even below this limit, iterative methods to construct initial data may converge to solutions on either one of the two branches, depending on the starting guess, leading to potentially inconsistent physical results in the evolution.

The scatterings between cold atoms, molecules, and ions provide an important playground for exploring cold physics and chemistry. Quantum scattering theory is essential to understanding the quantum mechanical nature of reactive and non-reactive scattering processes, providing comprehensive information on the dynamic scattering event. At low collision energies, the time-independent method is commonly used, but it suffers from a steep-scaling law with respect to the problem size. The present work develops the time-dependent wave packet method for non-reactive scatterings at low collision energies by constructing initial wave packets that are free of opposite momentum components, which otherwise emerge from the extended momentum range intrinsic to the finite Gaussian-shaped initial wave packet. These components have no influence on the reactive channel but interfere with the non-reactive elastic parts after being reflected by the barrier. Three schemes-the direct cut, the smooth elimination, and the direct construction-are proposed to alleviate the influence of the opposite components, and they are applied to three examples, a one-dimensional toy model, the H + H2 system, and the O + OH system, to demonstrate their performance in non-reactive scatterings at low collision energies.

We examine recent claims that "non-Newtonian" arithmetic and calculus topple Bell's theorem. Our basic point regarding such a claim is straightforward: the expectation functional used in those papers is linear only with respect to the deformed sum ⊕, not the ordinary +. Consequently, the familiar Clauser–Horne and Clauser–Horne–Shimony–Holt derivations — which lean on linearity under ordinary addition — do not apply. Within a single arithmetic level, a Bell-type analogue can be formulated if the outcomes and expectation values are defined in that level and satisfy linearity with respect to the level's addition ⊕; however, the standard Clauser–Horne/Clauser–Horne–Shimony–Holt proof for "+" is inapplicable. The eye-catching "beyond-Tsirelson" effects show up only when levels are mixed — thus, one computes with standard rules on quantities defined in a deformed calculus, producing out-of-range aggregates (e.g., totals exceeding one) rather than single-event probabilities. The touted "relativity of observed probabilities" also splices together two different moves, conditioning on a restricted sample space versus pushing everything through a scalar remapping. A simple horizon toy model already shows that there is no single-valued remapping that accomplishes this globally. The analogy with Einstein velocity addition helps a little in one dimension; in three dimensions, it collapses. There, the composition is non-commutative and non-associative, and the right language is a gyrogroup structure, not a pullback of ordinary addition. Bringing in Lambare's measurement-independence critique, we further argue that Czachor's reply (built from a hand-tuned bijection and a non-additive integral) addresses neither that objection nor Bell's own premises. In short, the program amounts to a representational re-encoding, not a counterexample of local hidden-variable.

Multicellular organisms utilize epithelial folding to achieve remarkable three-dimensional forms. During embryonic development, stereotypical epithelial folds emerge from underlying active cellular and molecular processes including cell shape change and differential cell growth. However, the origin of epithelial folding in early animals and how folding may be harnessed in synthetic systems remain open questions. Here, we identify a modality of behavior-induced epithelial folding and unfolding arising from cilia-substrate adhesion and ciliary walking in the basal animal Trichoplax adhaerens (phylum Placozoa). We show that T. adhaerens is capable of exhibiting dynamic nonstereotyped folding states, providing a 3D perspective to an organism previously only characterized in its 2D state. We correlate these folding states to local substrate geometry, revealing that the animal conforms to available substrate surface area, promoting the maintenance of a folded state. Using 4D fluorescence light sheet microscopy, we characterize fold geometry, curvature evolution during unfolding, and the nonstereotypy of unfolding behavior. Through repeated unfolding trials, we reveal the robustness and timescales associated with unfolding behavior and employ scaling analysis and toy model simulations to establish how collective ciliary activity can robustly drive unfolding. In this way, despite lacking any folding-unfolding "pathway," transitions between folding and unfolding states emerge as a function of the animal's environment and motility. Our work reveals a remarkable behavior exhibited by a brainless, nerveless animal, and demonstrates the capacity for 3D-2D transitions in folding epithelial sheets using ciliary activity.

Blind source separation (BSS) plays a pivotal role in modern astrophysics by enabling the extraction of scientifically meaningful signals from multi-frequency observations. Traditional BSS methods, such as those that rely on fixed wavelet dictionaries, enforce sparsity during component separation but can fall short when faced with the inherent complexity of real astrophysical signals. In this work, we introduce the learnlet component separator (LCS), a novel BSS framework that bridges classical sparsity-based techniques and modern deep learning. LCS utilises the learnlet transform -- a structured convolutional neural network designed to serve as a learned, wavelet-like multi-scale representation. This hybrid design preserves the interpretability and sparsity-promoting properties of wavelets while gaining the adaptability and expressiveness of learned models. The LCS algorithm integrates this learned sparse representation into an iterative source separation process, enabling the effective decomposition of multi-channel observations. While conceptually inspired by sparse BSS methods, LCS introduces a learned representation layer that significantly departs from classical fixed-basis assumptions. We evaluated LCS on both synthetic and real datasets and in this paper demonstrate its superior separation performance compared to state-of-the-art methods (average gain of about 5 dB on toy model examples). Our results highlight the potential of hybrid approaches that combine signal processing priors with deep learning to address the challenges of next-generation cosmological experiments.

Abstract The history of astronomical discovery shows that many of the most detectable phenomena, especially detection firsts, are not typical members of their broader class, but rather rare, extreme cases with disproportionately large observational signatures. Motivated by this, we propose the Eschatian Hypothesis: that the first confirmed detection of an extraterrestrial technological civilization is most likely to be an atypical example, one that is unusually “loud” (i.e., producing an anomalously strong technosignature), and plausibly in a transitory, unstable, or even terminal phase. Using a toy model, we derive conditions under which such loud civilizations dominate detections, finding for example that if a society is loud for only 10 −6 of its lifetime, it must emit ≳1% of its total observable energy budget during that phase to outrun quieter populations. The hypothesis naturally motivates agnostic anomaly searches in wide-field, multi-channel, continuous surveys as a practical strategy for a first detection of extraterrestrial technology.

The author analyzes modelling as a method of research inadequately developed in domestic legal studies yet widespread in the United Kingdom and the United States for that purpose. It proves a considerable heuristic potential of modelling for legal science in the context of digital change, with legal regulation based on predicting and assessing the implications and risks of rule-making as a substitute for reactive approach. It is pointed out a legal system analysis can be well-served not only by realistic models based on empirical data, but also by abstract semantic models employing the idealization method and deliberate distortion of simulated system’s qualities. The article identifies core methodological issues to be addressed for an adequate choice of models relevant to the specific research objective. It analyzes the typology of scientific models proposed by R. Frigg and S. Hartmann based on the target object’s representation type and justifies its applicability to legal studies for analysis of constraints of specific legal system models and their construction principles. The essential types of scientific models and their conceptual features are showcased by key papers of modern British and American legal science, with a focus on those widespread in analytical jurisprudence for building comprehensive theory of law and order. These include analogical models (H. Hart, R. Dworkin) designed to analyze the essential qualities of the legal system; idealized models (J. Austin, H. Kelsen) disregarding exogenous social factors that obstruct an analysis of law, and toy models (J. Bentham, L. Fuller) which use deliberately false system assumptions and exaggerate its specific qualities to analyze theoretic foundations. It is noted that modelling is crucial for analytical philosophy to identify essential qualities of law and reveal the internal logic of normative systems. While for each model type under study the article identifies methodological constraints inherent in interpretation of findings, it is concluded that such constraints should be treated with care and that methodological design is crucial for theoretic studies of law.

Abstract For an interferometric array, an image of the sky can be synthesized from interferometric visibilities, which are the cross-correlations of the received electric voltages of pairs of array elements. However, to search for transient targets such as fast radio bursts (FRBs), it is more convenient to use the beam-forming technique, where the real-time voltage outputs of the array elements are used to generate data streams (beams) which are sensitive to a specific direction. This is usually achieved by a weighted sum of the array element voltages, with the complex weight adjusted so that all outputs have the same phase for that direction. Alternatively, beams can also be formed from the weighted sum of the short time averaged correlation (visibility) data. We will call these two approaches the electric voltage beam forming (EBF) and cross-correlation beam forming (XBF), respectively. All beams formed with the EBF can also be formed by the XBF method, but the latter can also generate beams which cannot be generated by the former. We discuss the properties of these two kinds of beams, and the amount of computation Required in each case. For an array with large number of elements, the XBF would require much more computation resources, although this is partly compensated by the fact that it allows integration over time. We study the impact of cross-coupling between array elements on the beamforming, first using a toy model, then for the case of the Tianlai Cylinder Pathfinder Array. In both cases, we find that the impact of the cross-coupling on the beam profile is relatively small. The understanding gained in this study is helpful in designing and understanding the beam-forming FRB digital backend for compact arrays such as the Tianlai array.

Abstract We present the first ab initio lattice calculation of neutron–alpha ( n – α ) scattering using nuclear lattice effective field theory with chiral interactions at next-to-next-to-next-to-leading order (N3LO). Building on the high-fidelity chiral Hamiltonian introduced in Elhatisari et al (2024 Nature 630 59), we compute scattering phase shifts in the S - and P -wave channels using the Lüscher finite-volume method. Our results demonstrate excellent agreement with empirical R -matrix phase shifts in the 2 S 1/2 and 2 P 3/2 channels, while revealing persistent discrepancies in the 2 P 1/2 channel for neutron energies above 5 MeV. To systematically investigate these discrepancies, we construct and analyze a simplified neutron–alpha toy model, demonstrating that these discrepancies are not due to the use of the Lüscher finite-volume method. Additionally, we revisit our three-nucleon (3N) force fitting procedure, explicitly incorporating neutron–alpha scattering data through comprehensive Markov Chain Monte Carlo sampling. This analysis confirms the stability of nuclear binding-energy predictions and highlights the need for further refinements in the lattice N3LO three-nucleon forces to fully describe neutron–alpha scattering in the challenging 2 P 1/2 channel.

Abstract Understanding the coronal geometry in different states of black hole X-ray binaries is important for more accurate modeling of the system. However, it is difficult to distinguish different geometries by fitting the observed Comptonization spectra. In this work, we use the Monte Carlo ray-tracing code MONK to simulate the spectra for three simple corona toy models widely proposed in observational studies: sandwich, spherical, and lamppost, varying their optical depth and size (height). By fitting the simulated NuSTAR observations with the simplcut*kerrbb model, we infer the possible parameter space for the hard state and soft state of different coronal geometries. The influence of the disk inclination angle, black hole spin, and coronal temperature is discussed. We find that in the lamppost model, if we exclude the case of a very extended corona, the disk emission is always dominant, making lamppost geometry incompatible with the hard state. While the sandwich and spherical models can produce similar spectra in both the hard and soft states, the simulated IXPE polarimetric spectra show the potential to break this degeneracy. Geometrical effects arising from the limited size of the corona become evident in lower-spin black holes and affect the spectral fitting, where the larger ISCO reduces the corona coverage of the inner disk.

Abstract Mergers are believed to play a pivotal role in galaxy evolution, and measuring the galaxy merger fraction is a longstanding goal of both observational and theoretical studies. In this work, we extend the consideration of the merger fraction from the standard measure of binary mergers, namely those comprising two merging galaxies, to multiple mergers, namely mergers involving three or more galaxies. We use the Illustris and IllustrisTNG cosmological hydrodynamical simulations to provide a theoretical prediction for the fraction of galaxy systems that are involved in a multiple merger as a function of various parameters, with a focus on the relationship between the multiple merger fraction f m and the total merger fraction f t . We generally find that binary mergers dominate the total fraction and that f m ≈ ( 0.5 − 0.7 ) f t 5 / 3 , a prediction that can be tested observationally. We further compare the empirical simulation results with toy models where mergers occur, on the evolution timeline of a galaxy, either at constant intervals or as a Poisson process at a constant rate. From these comparisons, where the toy models typically produce lower multiple merger fractions, we conclude that in cosmological simulations, mergers are more strongly clustered in time than in these toy scenarios, likely reflecting the hierarchical nature of cosmological structure formation.

Context. The electromagnetic detection of circumbinary disks around pre-merger binary black holes (BBHs) relies on theoretical predictions. These are generally obtained through expensive numerical simulations, but simple or fast toy models are lacking to unleash the potential of these theoretical advances for observational purposes. Aims. We present a simple toy model for computing the electromagnetic variability of circumbinary disks around circular-orbit BBHs at relativistic separations. We focus on the effect of disk nonaxisymmetries. Methods. We assumed that the disk is threaded by spiral arms and hosts a hotspot linked to an overdense structure (the lump) that is preferably reported in binaries of a close to equal mass. We built a simple temperature distribution and estimated its thermal emission, perceived by a distant observer, via a ray-tracing code in a BBH approximate metric. We propose a toy model that reproduces the main light-curve features and show that it is consistent with 2D general relativistic hydrodynamical simulations under the assumption of compressional heating and expansional cooling, except for purely dynamical effects such as the binary-lump beat. Results. The light curve exhibits a main modulation at the lump period (i.e., a few times the orbital period) due to the relativistic Doppler effect, and a shorter modulation at the orbit-like period due to spiral arms or the beat. These are more prominent in the optical/UV band for a total binary mass M = 10 4 − 10 M ⊙ , where the disk energy spectrum peaks. For M = 10 9 M ⊙ , a lump modulation with an amplitude of 4% is detectable with the Vera Rubin Observatory after six months of observations up to z = 0.5. Conclusions. We proposed a new simple toy model that can be used, for instance, to test the compatibility of the periodicity of BBH candidate sources with a circumbinary disk origin.

Parkinson's disease (PD) is a progressive neurodegenerative disorder characterized by motor and non-motor symptoms that impair gait and daily activities. In this study, we propose a novel method based on recurrence plot and recurrence triangle (RT) patterns to classify PD patients and healthy controls. We analyze a toy model using the Rössler attractor as well as real-world walking datasets. The RT-based approach showed exceptional performance, achieving almost 100% classification accuracy on the toy model and real-world datasets. Statistical analysis revealed that specific RT patterns-type 13, type 20, and type 51 for triangle size L=4, and type 787 and type 819 for triangle size L=5-occur more frequently in healthy individuals and reflect structured, rhythmic, and adaptive gait dynamics. In contrast, triangle types 1, 60, and 64 are observed more frequently in PD patients, capturing irregular motion, fluctuations, or slow behavior. These findings demonstrate that RT patterns provide interpretable features to distinguish a healthy and pathological gait. This study highlights the potential of RT-based analysis as an accurate and interpretable tool for early detection and diagnosis of PD, paving the way for future clinical applications.

Abstract An earthquake record is the convolution of source radiation, path propagation and site effects, and instrument response. Isolating the source component requires solving an ill‐posed inverse problem. Whether the instability of inferred source parameters arises from varying properties of the source, or from approximations we introduce in solving the problem, remains an open question. Such approximations often derive from limited knowledge of the forward problem. The Empirical Green's function (EGF) approach offers a partial remedy, by approximating the forward response of large events using the records of smaller events. The choice of the best small event drastically influences the properties estimated for the larger earthquake. Discriminating variability in source properties from epistemic uncertainties, stemming from the forward problem or other modeling assumptions, requires us to reliably account for, and propagate, any bias or trade‐off introduced in the problem. We propose a Bayesian inversion framework that aims at providing reliable and probabilistic estimates of source parameters (here, for the source‐time function or STF), and their posterior uncertainty, in the time domain. We jointly solve for the best EGF using one or a few small events as prior EGF. Our approach expands on DeepGEM, an unsupervised generalized expectation‐maximization framework for tomography. We demonstrate, with toy models and various applications to mainshocks of ranging from 4 to 6.3, the potential of DeepGEM‐EGF to disentangle the variability of the seismic source from biases introduced by modeling assumptions.

This work proposes a geometric-statistical reinterpretation of the dark sector, grounded in a discrete spacetime framework composed of non-material spatial units termed hodons. Unlike particle-based dark matter models, hodons are kinematically inert and possess ultra-light effective mass derived from vacuum energy density and holographic volume bounds. We introduce a covariant scalar field representing local hodon density and derive an entropy-driven evolution equation consistent with causal structure and general relativity. The resulting stress-energy contribution from hodon fluctuations yields gravitational clumpiness without invoking new particles or modified gravity. A virial-based toy model demonstrates that baryonic matter surrounded by hodons forms stable, cored halo profiles, consistent with galactic rotation curves and low-mass halo observations. The framework naturally suppresses small-scale structure via spatial uncertainty relations, aligning with constraints from the Lyman-α forest and weak lensing. By integrating Bousso's covariant entropy bound and distinguishing between strong and weak holography, we situate the model within a broader epistemological context. These results suggest that dark sector phenomenology may emerge from the statistical geometry of space itself, offering a falsifiable alternative to particle dark matter.