We propose Scalable Quantum Molecular Generation (SQMG), a variational quantum-circuit for sampling molecular graphs using chemical priors on atoms and bonds. SQMG assigns a fixed 3-qubit register to each heavy atom and reuses a single 2-qubit bond register to generate bonds sequentially, yielding an ''atom no-reuse, bond reuse'' architecture with linear qubit scaling. Measurement results are mapped to molecular graphs via lightweight classical decoding with structural constraints. In CUDA-Q, we benchmark the state-vector simulation (CPU/GPU) and the tensor-network simulation (GPU). At $N=8$ heavy atoms, the state-vector simulator (GPU) and the tensor-network simulator (GPU) achieve speeds of up to $4.5\times 10^{4}$ and $2.2\times 10^{3}$ over the state-vector (CPU) baseline, respectively. Crucially, tensor-network simulation extends exact simulation to $N=40$ heavy atoms, where state-vector methods become memory-limited. For training, Bayesian optimization outperforms COBYLA on a Validity$\times$Uniqueness objective, and the same architecture supports \textit{de novo} generation, scaffold decoration, and linker design. Overall, SQMG provides a scalable, reproducible testbed for evaluating accelerated tensor-network simulation and future quantum molecular generation algorithms.

We study transient entanglement generation between two quantum nodes coupled through a chiral one-dimensional channel. In an emitter-only Born-Markov description, we show that continuous driving and an initial ground state can raise the maximum transient concurrence above the undriven $2/e$ benchmark associated with the effectively single-excitation model. We then consider a more microscopic XX spin-chain channel with triangular plaquette couplings and compare a nonsecular time-convolutionless master equation (TCL-ME) with matrix-product-state (MPS) simulations. In the optimal driven regime, the nonsecular TCL-2 treatment reproduces the concurrence envelope and first transient peak qualitatively, while the remaining discrepancy is mainly attributable to beyond-Born system-bath correlations. The enhancement is traced to the failure of the secular approximation under strong driving, where nearby dressed transitions are not well separated on the dissipative timescale and nonsecular terms mix dressed-state coherences. Finally, we examine within TCL-2 the sensitivity of the protocol to positional disorder, imperfect chirality, and loss into nonguided modes. These results clarify when the familiar $2/e$ limitation ceases to apply and separate the roles of secular breakdown, Born-factorization error, and reduced-state memory in driven chiral entanglement generation; we believe that our study contributes to one of the first studies where the breakdown of the secular approximation is useful rather than detrimental.

The new family of superconducting nickelates centered around La$_{3}$Ni$_{2}$O$_{7}$ possesses attractive features, such as the high transition temperature and the presence of an antiferromagnetic ground state at ambient pressure, suggesting an unconventional pairing mechanism. In the nonsuperconducting state, the possibility of different density-wave orders with opposite pressure dependencies is discussed, whose relationships and microscopic origins are largely unknown. However, sample-quality issues, such as impurity-phase formation or oxygen vacancies, impede the progress in the field. Here, we employ high-pressure synthesis and hydrostatic high-pressure transport techniques to investigate bilayer nickelates with controlled band width and filling, and perform a systematic study on their impact on the superconductivity and other characteristic properties. While increasing the tilting of the NiO$_6$ octahedra shifts the superconducting phase to higher pressure, simultaneous hole doping reverts this trend. We also observe up to three distinct anomalies in the nonsuperconducting state which are possibly related to density-wave formation.

We propose a definition of an Euler characteristic for unbounded chain complexes by taking the (usual) Euler characteristics of successively longer parts of the complex, weighted inversely proportional to the length, and passing to the limit. This amounts to taking the limit of the sequence of ranks of homology modules with alternating signs in the sense of the Hölder summation method. We establish the structure of a category with cofibrations and weak equivalences on unbounded complexes for which the infinite Euler characteristic is defined, and show that its Grothendieck group is unusually large (viz., uncountable).

Quantum simulation with controllable many-body platforms offers a powerful route to exploring complex phases and dynamics that are difficult to access in natural materials. Among these, topological spin textures such as skyrmions are central to modern condensed-matter physics and play a key role in chiral quantum many-body systems. Their controlled realisation in large, programmable quantum platforms, however, remains an outstanding challenge. Here, we report deterministic generation and site-resolved reconstruction of topological spin textures in a two-dimensional crystal of more than 150 trapped ions. Using globally applied spin-dependent forces, we generate skyrmion configurations and reconstruct the full vector spin field with single-ion resolution, obtaining a winding number of 0.99$\pm$0.02 and a mean local fidelity of 0.87$\pm$0.04. In addition, we implement single-ion-resolved control to deterministically prepare domain-wall states, extending our approach to a broader class of non-uniform spin textures. These results establish trapped-ion crystals as a platform for engineering complex spin textures and open the door to exploring topology-dependent nonequilibrium dynamics in long-range interacting quantum systems.

We study bipartite entanglement harvesting from the quantum vacuum of a massless scalar field between two subsystems, each composed of a finite number of Unruh-DeWitt detectors. Using perturbation theory, we show that the leading-order negativity is fully determined by a submatrix of the reduced density matrix, with the submatrix dimension scaling only linearly with the number of detectors. Within this framework, we analyze how the detectors' spatial arrangement influences harvesting. For all three-detector configurations and several symmetric four-detector configurations, we derive analytic expressions for the negativity and identify the configurations that maximize it. For a linear chain, we find that the harvested entanglement scales linearly with the number of detectors. These results clarify how to arrange multiple detectors to optimize harvesting and show that increasing their number broadens the ranges of energy gaps and separations over which entanglement can be extracted from the field.

Transient noise artifacts, commonly referred to as glitches, pose a major challenge to parameter inference for space-based gravitational-wave (GW) observations. We develop a glitch-robust amortized inference framework for massive black hole binaries in the Taiji detector configuration by combining conditional normalizing flows, a time-frequency multimodal fusion encoder, and contrastive learning. To enable large-scale training on contaminated data, we further introduce a neural glitch generator that produces high-fidelity synthetic transients at substantially reduced computational cost. Systematic experiments show that, under glitch contamination, the proposed method yields more accurate and better-calibrated posteriors than a conventional Markov Chain Monte Carlo baseline. In ablation studies, the full time-frequency model with contrastive learning performs best overall and remains robust to variations in glitch duration and merger-relative timing. We further show that standard coverage diagnostics alone are insufficient to fully assess posterior fidelity. We therefore complement them with the continuous ranked probability score, which provides a stricter assessment of global distributional agreement in non-ideal GW data. Taken together, these results establish deep-learning-based amortized inference as a promising framework for fast and robust Bayesian parameter estimation in future space-based GW observations.

Direct observational constraints on the earliest, stellar-wind-dominated phases of galactic outflows remain scarce. We present medium-resolution VLT/X-shooter spectroscopy of six Type I superluminous supernova (SLSN-I) host galaxies at z = 0.15-0.51, exploiting the bright SLSN continua as single, down-the-barrel probes of the host interstellar medium. From nebular emission lines we derive dust-corrected star-formation rates as low as 0.06-0.44 Msun yr$^{-1}$, and gas-phase metallicities in the extremely metal-poor regime (less than nine percent solar). Moreover, all hosts exhibit narrow, blueshifted Mg II 2796, 2803 absorption, indicative of the presence of low-ionization outflows along the line of sight. Voigt modeling of the Mg II absorption yields maximum outflow velocities of $v_{max}$ = 37-104 km s$^{-1}$, placing these galaxies systematically below the empirical $v_{max}$-SFR relations for more evolved galaxies of similar SFR. Given the short lifetimes of the SLSN massive progenitors, we argue that these outflows must originate from preceding stellar wind episodes. Assuming a constant-velocity outflow over 3 Myr and spherical symmetry, we infer wind masses M$_{wind}$ = (0.02-1.0) $\times 10^6$ Msun and mass-outflow rates $\dot{M}_{wind} = 0.01-0.33$ Msun yr$^{-1}$, corresponding to mass-loading factors $η<1$. These results indicate that, during the first few Myr of a burst, stellar winds and radiation pressure alone drive slow and weak outflows in low-mass systems, prior to the onset of dominant supernova feedback. Our work provides one of the first empirical constraints on early feedback phases relevant for high-redshift galaxies, and for time-dependent implementations of stellar feedback in galaxy formation simulations.

Structural disorder in the magnetically frustrated pyrochlore system leads to intriguing magnetic states. We present the thermodynamic behavior and short range magnetic correlations in Tb$_{2}$Zr$_{2}$O$_{7}$ and Tb$_{2}$Zr$_{1.5}$Ti$_{0.5}$O$_{7}$ compounds. The parent compound Tb$_{2}$Zr$_{2}$O$_{7}$ has defect fluorite structure, which evolves toward the pyrochlore phase on Ti doping at Zr site. There is no long range magnetic order down to 0.4 K, and a magnetic field dependent spin freezing evolves below 1.25 K and 1.05 K for the parent and doped compounds, respectively. The ac susceptibility measurements indicate slow spin relaxation process below 20 K in these compounds. Inelastic neutron scattering reveals broad diffuse scattering, indicative of short range correlations at low temperature, owing to local structural distortions and persistent spin fluctuations. These results suggest a correlated, disorder influenced magnetic state in Tb$_{2}$Zr$_{2}$O$_{7}$, Tb$_{2}$Zr$_{1.5}$Ti$_{0.5}$O$_{7}$ compounds.

We propose a matrix zonotope perturbation framework that leverages matrix perturbation theory to characterize how noise-induced distortions alter the dynamics within sets of models. The framework derives interpretable Cai-Zhang bounds for matrix zonotopes (MZs) and extends them to constrained matrix zonotopes (CMZs). Motivated by this analysis and the computational burden of CMZ-based reachable-set propagation, we introduce a coefficient-space approximation in which the constrained coefficient space of the CMZ is over-approximated by an unconstrained zonotope. Replacing CMZ-constrained-zonotope (CZ) products with unconstrained MZ-zonotope multiplication yields a simpler and more scalable reachable-set update. Experimental results demonstrate that the proposed method is substantially faster than the standard CMZ approach while producing reachable sets that are less conservative than those obtained with existing MZ-based methods, advancing practical, accurate, and real-time data-driven reachability analysis.

The strong dipole-dipole interaction of trapped Rydberg ions offers the possibility of sub-microsecond entanglement gates. For example a two-qubit Control-Phase gate in 88 Sr + ions can be realized, by simultaneous excitation to the Rydberg states via stimulated Raman adiabatic passage (STIRAP) with simultaneous microwave induced dipole-dipole interaction. We show that this excitation protocol distorts the dark-state of the STIRAP stage and is prone to decay from the intermediate state. Here, we propose a novel pulse ordering, in which the STIRAP and the microwave dressing of the Rydberg states occurs in separate stages, preventing mutual interference effects that are detrimental to the gate fidelity. We show that, for experimentally feasible parameters, the proposed excitation scheme can achieve a fidelity of 99.93%, surpassing the experimentally demonstrated gate. In addition, we demonstrate a non-adiabatic speed-up to 400 ns by employing asymmetric pulse shapes in the STIRAP stage. The entangling phase is then controlled solely through the interaction strength by nonresonant asymmetric chirping of the microwave field.

Ultrashort-pulse laser ablation of metals near damage threshold is governed by homogeneous spallation, in which tensile unloading releases a nanometre-thin liquid film whose optical signatures are temporally evolving concentric Newton rings in pump--probe experiments. This well-established picture rests almost exclusively on single-pulse results obtained on ideally flat surfaces, yet application-oriented processing invariably operates in a multi-pulse regime in which each pulse irradiates a surface progressively modified by preceding pulses. Whether homogeneous spallation persists under these conditions has remained an open question. Here we resolve this question using time-resolved pump-probe interferometry applied pulse by pulse to austenitic stainless steel. We show that homogeneous spallation dominates the first pulse, while its contribution is strongly reduced for the second pulse. By the third pulse, Newton rings vanish and sustained surface bulging collapses, with the optical transients fully saturating into a phase-explosion-like signature by the fourth pulse. Fourier-domain coherence analysis rules out roughness-induced decoherence as an optical artefact. Four independent observables, spanning time-resolved and final-state measurements, converge on the same transition after three to four pulses. Spallation-layer formation, widely invoked to explain ultrashort-pulse ablation of metals, is thus a single-pulse phenomenon rather than a multi-pulse ablation mechanism.

In this paper, we consider the design of Model Predictive Control (MPC) algorithms based on Mamba neural networks. Mamba is a neural network architecture capable of sub-quadratic computational scaling in sequence length with state-of-the-art modeling capabilities. We provide a consistent and complete mathematical description of the Mamba neural network is provided. Then, adjustments and optimizations are made to construct a decoder-only Mamba multi-step predictor for MPC and an input-output formulation is given for sequence-to-sequence modeling of dynamical systems. The performance of Mamba-MPC is evaluated on several numerical examples and compared to a Long-Short-Term-Memory based MPC (LSTM-MPC) equivalent. First, a Single-Input-Single-Output (SISO) Van der Pol oscillator is considered, where stability, reference tracking, and noise robustness are evaluated. Then, a MIMO Four Tank setup is introduced where Multiple-Input-Multiple-Output (MIMO) reference tracking is evaluated. Lastly, Mamba-MPC is implemented on a physical Quanser Aero2 setup for closed-loop reference tracking. The results demonstrate that Mamba-MPC is able to stabilize and track a reference for SISO and MIMO systems, both in simulation and on a physical setup. Moreover, Mamba-MPC consistently outperforms LSTM-MPC in predictive control and is significantly computationally faster.

We build solutions to Kac's particle system and show that their empirical measures converge to the solution of the space-homogeneous Boltzmann equation in the regime of very soft potentials. This proves propagation of chaos for the last class of kernels for which it was still open. The proof relies on new estimates on the dissipation of the Fisher information along the Boltzmann equation, which allow us to control the strong singularities of the system. These estimates are obtained thanks to a new inequality related to the fractional heat flow on the sphere, that might be of independent interest.

This paper provides a new proof of the $p$-adic Gross--Zagier formula for the $p$-adic $L$-function associated with the base change of a normalised cuspidal eigen-newform $f$ of weight $k \geq 2$ (and families of such) to an imaginary quadratic field $K$. Our results encompass both the classical $p$-ordinary cases and non-ordinary scenarios, including new cases where $k > 2$ and $\mathrm{ord}_p(a_p(f)) > 0$. Unlike the traditional approach of comparing geometric and analytic kernels, we employ a ``wall-crossing'' strategy centred on the BDP formula and the theory of Beilinson--Flach elements.

We derive the asymptotic first passage time (FPT) distribution for space-dependent variable-order time-fractional diffusion, where the fractional exponent $α(x)$ varies with position. For any sufficiently smooth $α(x)$ on a finite domain with absorbing and reflecting boundaries, we show that the survival probability decays as $Ψ(t)\sim C\,t^{-α_*}/(\ln t)^ν$, where $α_*$ is the minimum value of the fractional exponent and $ν$ is determined by the location and shape of the minimum. For a constant fractional exponent $ν=0$ and this provides a theoretical prediction that can identify spatially heterogeneous anomalous transport in experiments. We validate the theory against exact Laplace-space solutions and Monte Carlo simulations for linear and nonlinear profiles of $α(x)$.

Cosmological parameter inference from galaxy clustering relies critically on accurate estimates of the covariance and precision matrices. These are often obtained from a limited number of mock catalogs, introducing noise and bias in the precision matrix when the data-vector dimension becomes comparable to the number of available realizations. We present the first application of the Rotational Invariant Estimator (RIE) to the large-scale clustering of galaxies, benchmarking it against the standard sample covariance and the non-linear shrinkage estimator NERCOME for both the two-point correlation function (2PCF) and power spectrum. Using controlled synthetic data sets with analytically known covariance matrices, we estimate the covariance with all three methods across a range of mock-to-dimension ratios $q = N/D$ and data-vector sizes $D$. We then perform Bayesian inference with an EFT-based model and quantify each estimator through the Figure of Bias (FoB) and Figure of Merit (FoM). After correction for finite-$N$ effects, the sample covariance recovers unbiased average uncertainty volumes but suffers from growing best-fit scatter and bias at small $q$ due to the Dodelson--Schneider effect. Both NERCOME and RIE substantially reduce these stochastic shifts; however, the uncertainties they assign are probe-dependent. In configuration space, both estimators can yield overly tight constraints, with a bias that grows with $D$. In Fourier space, RIE delivers markedly improved best-fit stability with only mild FoM bias, whereas NERCOME tends to overestimate the constraining power. Among the estimators tested, RIE emerges as the most effective at stabilizing best-fit recovery, particularly in Fourier space, where it closely reproduces the reference posteriors even when the number of mocks barely exceeds the data-vector dimension.

Let $F_n$, $W_n$, and $\widehat{K}_n$ be the graphs obtained by joining a vertex to $n$ independent edges, a cycle and a path of order $n-1$, respectively. In this paper, we give new bounds for the Ramsey numbers $R(F_n,F_m)$ and $R(W_n,W_n)$, which improve those due to Chen, Yu, and Zhao [EJC, 2021] and Mao, Wang, Magnant, and Schiermeyer [G&C, 2022], respectively, and establish lower and upper bounds for $R(\widehat{K}_n,\widehat{K}_n)$. Moreover, we present a blow-up technique to establish some new lower bounds for the Ramsey numbers of wheels versus cliques.

Machine learning potentials (MLPs) achieve near first-principles accuracy but often fail for atomic environments outside the training distribution. Active learning can mitigate this limitation; however, its application to large-scale simulations is hindered by the prohibitive cost of labeling entire configurations. Here, we develop a D-optimality-driven active learning framework for the neuroevolution potential (NEP) implemented within the GPUMD package, named NEPMaker. Extrapolative atomic environments are identified on-the-fly and embedded into locally periodic structures, where boundary atoms are optimized to remain close to the training distribution. This strategy enables large-scale simulations to directly contribute to dataset construction, significantly reducing extrapolation errors while improving model robustness and transferability. The proposed framework provides a scalable route for constructing reliable machine learning potentials in complex materials systems, including those involving defects, interfaces, and phase transitions.

We investigate one-loop corrections from torsion-induced four-fermion interactions to inflaton three-body decay and their impact on the associated stochastic gravitational-wave signal. We find a pronounced asymmetry in the dependence on the renormalization scale $u$. While the enhancement of the gravitational-wave spectrum remains modest, not exceeding roughly a factor of order unity for representative inflaton masses well below the Planck scale within the perturbative regime, the suppression can be much stronger, reaching up to two orders of magnitude, corresponding to reductions at the percent level. These results imply that loop corrections, particularly fermionic self-interactions, can significantly reduce the predicted gravitational-wave signal in models based on tree-level analyses. This suppression may shift the signal outside the sensitivity range of future observations and should therefore be taken into account in realistic phenomenological studies.

We developed a machine learning interatomic potential (MLIP) for Ge-rich GeSbTe alloys of interest for applications in phase change memories embedded in microcontrollers. The MLIP was generated by fitting with a neural network method a large database of energies and forces computed within density functional theory of elemental, binary, stoichiometric and non-stoichiometric ternary alloys in the Ge-Sb-Te phase diagram. The MLIP is demonstrated to be highly transferable to large regions of the phase diagram around the compositions included in the dataset. The MLIP is then exploited to simulate the crystallization with phase separation of three Ge-rich alloys on the Ge-Sb$_2$Te$_3$ and Ge- Ge$_2$Sb$_2$Te$_5$ tie-lines that correspond to the set process of the memory cell. The transformation on the ns time scale and at 600 K, comparable to the operation conditions of the memory, yields crystalline cubic GeTe slightly Sb-doped and amorphous GeSb and Ge. These metastable phases differ from the thermodynamically stable products and form due to kinetics effects on the short time span of the set operation in phase change memories.

The invariance principle, through which the steady-state behavior of nonlinear systems was introduced by Isidori and Byrnes, is leveraged in this article to bring forth a unifying characterization of the frequency response of nonlinear systems. We show that, for systems under nonlinear periodic excitations, the frequency response can still be defined as a complex-valued function in a phasor form. However, together with suitable notions of gain and phase functions, we show the existence of another function that completes the frequency response and allows quantifying the distortion introduced by the system in the steady-state output. This nonlinear characterization enabled the representation over input frequency and amplitude of the gain, phase, and distortion produced by the system, via a nonlinear enhancement of the Bode diagrams. This graphical representation of the frequency response is well-suited to performance analysis of a nonlinear system and, furthermore, allows for the formulation of the loop-shaping problem for nonlinear systems.

The Melnikov-Arnold integrals (MA-integrals) is a well-known instrument used to measure the splitting of separatrices in Hamiltonian systems. In this article, we explore how calculation of MA-integrals can be used as well to estimate sizes of secondary resonances. Within the standard map model, we show how the newly developed MA-based procedure allows one to estimate the sizes of secondary resonances of any order (up to the order of the optimal normal form), without relying on the cumbersome traditional normalization procedure.

We present Basilic, a dedicated pipeline for Bayesian model selection and parameter estimation of short-duration gravitational-wave burst signals observable with ground-based detectors. Built on top of the bilby framework, Basilic combines modularity, pre-implemented burst models, and HTCondor integration to enable rapid, user-friendly analyses with minimal technical overhead. This work outlines the design philosophy, operational flow, and a set of example use cases demonstrating its scientific potential. As a case study, we also undertake an in-depth exploration of the comparison between a binary black hole merger and a cosmic string signal, through a parameter space exploration injection campaign. In addition to the well-known high-mass binary black-hole signal morphology degeneracy with cosmic string-like signals, we find that high anti-aligned component spins, even at moderate mass, can result in a similar degeneracy. Motivated by the likely low-SNR expected regime of possible future detections, we propose a data-driven study of model degeneracy, to be employed in the event of an inconclusive Bayes factor.

This paper studies local existence and the singularity formation of the solutions of the one-dimensional hyperbolic Navier-Stokes equations, in particular proving the gradient blow-up of the derivatives of the solutions. The underlying model introduces a relaxation mechanism that leads to hyperbolization, achieved both through a nonlinear Cattaneo law for heat conduction and through Maxwell-type constitutive relations for the stress tensor. Our main approach is to prove that the hyperbolic Navier-Stokes equations are indeed hyperbolic, and to prove that they possess two genuinely nonlinear eigenvalues, thereby establishing the blow-up of the gradient of the solution. In addition, we provide a derivation of the equation of state for the hyperbolic Navier-Stokes equations in the appendix.

Conventional quantum routing operates under the entrenched assumption that pathfinding is a prerequisite for routing. This classical-inspired routing model imposes a restricting design option, which prevents scaling the quantumness to the network functioning. In this paper, we proposed a novel entanglement-driven routing framework that exploits multipartite entanglement complementation for enabling simultaneous 1-hop connectivity among all non-adjacent source-destination pairs. This changes the notion of ``remoteness'' in the entanglement graph, activated by entanglement. We extend this framework to inter-domain quantum networks and design a polynomial-time algorithm. Such an algorithm allows to select and parallelize multiple requests, bypassing NP-complete path discovery. Performance analysis shows the proposed routing strategy achieves up to $60\%$ hop reduction, with the algorithm enabling efficient parallelism and strong scalability in inter-domain quantum networks.

Timelike orbits in curved spacetimes encode intrinsic information about the background geometry and serve as critical probes for investigating gravitational theories and source distributions. In this study, we investigate strictly closed timelike orbits within a Schwarzschild spacetime embedded in a Dehnen-type dark matter halo. By solving the geodesic equations, we identify various configurations of these closed orbits and simulate their corresponding gravitational waves and electromagnetic light curves. Our findings reveal that the morphology of closed orbits is primarily governed by the ratio of the azimuthal period to the radial period. Notably, dark matter halo parameters such as the core scale and density parameters exert a significant amplification effect on the orbital scale, which further induces a discernible phase lag in the gravitational wave signals. Furthermore, although certain orbital structures including the number of leaves remain challenging to distinguish via gravitational wave signals alone, they exhibit identifiable signatures in the characteristic peaks of light curves. These findings reveal the multi-messenger potential of closed orbits in bridging black hole environments and dark matter properties, providing theoretical guidance for future dark matter searches.

Vickrey's classic single-bottleneck departure time choice equilibrium model exhibits instability under many plausible day-to-day learning dynamics. Such instability is not observed in reality -- does this difference stem from the day-to-day dynamics or from one of the simplifying assumptions of the basic model? This paper explores a variant of the basic model with a continuous distribution of schedule delay parameters which we intuitively expect to have more favorable stability properties. To attain tractability we assume a monotonic relationship between earliness and lateness parameters. We first verify the existence and uniqueness of the equilibrium solution for this model. We then study a broad class of day-to-day dynamics satisfying local pressure and order preservation conditions. Our main contribution is a formal proof that, surprisingly, all such day-to-day dynamics in this context are unstable.

Quantitative modeling has become an essential tool in modern biophysics, driven by advances in both experimental techniques and theoretical frameworks. Powerful high-resolution techniques now provide detailed datasets spanning molecular to tissue scales, allowing to visualize cellular structures with unprecedented detail. In parallel, developments in soft and active matter physics have established a robust theoretical basis for describing biological systems. In this context, two main modeling paradigms have emerged: particle-based models, which explicitly represent discrete components and their interactions, and continuum models, which describe systems through spatially varying fields. We compare these approaches across biological scales, highlighting their respective strengths, limitations, and domains of applicability. To keep our discussion biologically relevant, we focus on five systems of fundamental importance: the cytoskeleton, membranes, chromatin, biomolecular condensates and tissues. With this Review, we thus aim to provide a framework for both theorists and experimentalists to select appropriate modeling strategies, and highlight future directions in biophysical modeling.

Analytic self-maps of the unit disc whose hyperbolic derivative is uniformly bounded by a constant smaller than one, are called contractive. We describe these maps in terms of their Aleksandrov-Clark measures and in terms of their inner-outer factorization. In addition, we show that contractive inner functions can be described in terms of a certain mixing property of its boundary values. We also present other results on the boundary behavior of contractive inner functions.