Papers

Representability determines when a two-particle reduced density matrix (2-RDM) corresponds to a physical quantum state, enabling many-particle quantum calculations with 2-RDMs rather than the wave function. In this Letter, we present a solution of the representability problem for quantum systems without particle-number conservation. The physically allowed set of 2-RDMs can be characterized from a geometrically `orthogonal' set, the polar cone. We derive explicit linear equations for the two-body operators in the polar cone – the intersection of the $p$-positive cone with the two-body operator space – to obtain a systematic hierarchy of representability conditions that do not depend on higher RDMs or the wave function. Moreover, by augmenting these conditions with the particle-number variance, we obtain a unified framework for treating both particle-number-conserving and nonconserving systems. We illustrate with a spin system and molecular H$_4$.

We investigate an analytic theory of force-dipole hydrodynamics in a viscous membrane coupled to an infinite surrounding fluid, focusing on quenched (orientation-fixed) dipoles. While the single-dipole flow exhibits the known Saffman crossover from a near-field $v\sim r^{-1}$ to a screened far-field $v\sim r^{-2}$, we show that this crossover induces a qualitatively new reorganization of dipole–dipole interactions. For two identical quenched dipoles, the near-field dynamics is exactly solvable and effectively one-dimensional, with a fixed line of centers and linear evolution of the squared separation. In the far field, the system remains integrable but becomes intrinsically two-dimensional, with coupled radial and angular dynamics and an exact first integral. For pullers, the angular dynamics drives alignment toward an attracting manifold, leading to universal late-time collapse $R\sim (t_c-t)^{1/3}$, in contrast to the near-field scaling $R\sim (t_c-t)^{1/2}$. The Saffman crossover thus reorganizes the Hamiltonian phase-space structure of dipolar interactions and produces a transition from effectively one-dimensional to fully coupled dynamics, providing a minimal framework for aggregation in viscous fluid membranes.

We study a dissipative extension of vortex-binary motion in a doubly periodic fluid domain. The underlying conservative system admits an exact integrable reduction to a single complex relative coordinate. Dissipation is introduced via a minimal rotated-velocity (mutual-friction) term, as motivated by finite-temperature superfluid dynamics, converting the Hamiltonian evolution into a mixed symplectic–gradient flow with monotonic energy decay for quantized vortices. In the local regime, the dissipative binary remains analytically solvable and admits closed-form solutions, with systematic corrections arising from the toroidal geometry. Equal same-sign vortices execute outward spiraling motion, while equal opposite-sign pairs (dipoles) undergo finite-time collapse in the planar limit. On the torus, however, the dipole orientation is no longer invariant: the geometry induces a slow angular drift, even in regimes where planar dynamics would preserve alignment. For unequal opposite-sign pairs, dissipation induces coupled contraction and rotation, leading to a finite-time nonlinear chirp characterized by $\dot{\omega}\propto\omega^2$, in contrast with electromagnetic and gravitational inspirals where $\dot{\omega}\propto \omega^{3}$ and $\dot{\omega}\propto \omega^{11/3}$. These results highlight the interplay between Hamiltonian structure, dissipation, and geometry in periodic fluid systems.

In this work, an LES model with finite-rate phase change and subgrid spray based on a high-resolution numerical scheme for multiphase multi-component simulations which satisfies interface equilibrium and phase immiscibility conditions is proposed. The multiphase model is based on a robust implementation of the four-equation multiphase model which assumes a strict subgrid equilibrium of pressure, temperature, and velocity. Critically, the equilibrium assumptions of the four-equation model provide large computational savings compared to modeling the full non-equilibrium multiphase system. To obtain predictive capabilities with these restrictive equilibrium assumptions, a new phase-confined form of the Eulerian $\Sigma$ spray model is proposed to predict subgrid interfacial surface area while avoiding unphysical leakage across interfaces. Additionally, an improved finite rate phase change model which is thermodynamically bounded by the equilibration of the Gibbs-free energy is coupled with the $\Sigma$ equation to model complex phase change regimes. The full modeling framework is validated using the Engine Combustion Network (ECN) Spray A case in non-evaporating and evaporating conditions and shows excellent agreement with experimental measurements.

To date, frequency-dependent squeezed light has been used to reduce quantum noise in interferometric gravitational-wave detectors by 6.1 dB (a factor of two). Future upgrades and detectors aim to both reduce quantum noise by 10 dB (a factor of three) and to increase the circulating power in the interferometer arm cavities. Achieving these goals will be extremely challenging due, in part, to the degradations to the squeezed state caused by mode mismatch between the internal interferometer optical cavities and between the auxiliary external cavities. It is therefore imperative to gain a detailed understanding of all sources of mismatch and to obtain experience in mitigating their effects in the current detectors in order to improve astrophysical sensitivity now and in the future. Two types of internal mismatch are identified which are due to the thermal aberrations generated when the test mass optics absorb a small fraction of the circulating arm power. It is found that the dynamics responsible for the degradations caused by the mismatch between the quadratic part of the wavefront of two modes has a characteristic low-pass frequency dependence while the dynamics of the mismatch due to all higher order thermal aberrations has a high-pass behavior. As a consequence, the two types of mismatch are predominantly responsible for different squeezing degradations—some of which are significant for the current detectors and some of which will only be important for future detectors with longer arms. The behavior of these two types of internal mismatch are described, and the implications for detector design, operation, and characterization are discussed.

Seminal works on animal collectives started proposing a diffusive model (overdamped) for the information transfer occurring in it \cite{Vicsek}. Afterwards, the introduction of self-rotational inertia brought into play an underdamped model able to better describe the information flux occurring in a real tuning flock event \cite{Atta}. That model was recently improved by adding nonlinear torques which allowed to match experiments \cite{cavagna2025}. The current work extends the latter model by adding active torques to a one-dimensional flock of boids (bird-like objects) while keeping key ingredients such as self-rotational inertia and nonlinearity. Those active torques are seen to enhance the system's information transfer speed and efficiency during a turning event, as well as rendering it a non-reciprocal status. The proposed internal active torques are motivated by the adaptive injection of rotational energy (active system) of birds in a real flock while turning. The continuum limit of the proposed model leads to a non-reciprocal modified Korteweg-de Vries (mKdV) equation with dissipation, whose structure allows the information transfer speed to be a function of the turning angular velocity. This feature occurs in real birds since under threat, birds turn faster and are required to get the information more rapid to keep cohesion.

Young interference is usually read as the gradual statistical accumulation of random detection events. Here we show that a time-reversed Young (TRY) geometry has a different statistical character: the fringe is not a marginal distribution of detector positions, but a conditional response indexed by a programmed source coordinate. With a fixed detector and a scanned source basis, the observable is an operational hybrid correlator between detector signal and source label. The resulting interference is deterministic at the response-function level, while noise enters only through estimation precision. We formulate this distinction using Fisher information, estimator variance, and noise scaling, clarifying why TRY naturally supports calibration, lock-in readout, null-fringe sensing, and source-plane superresolution.

The Peyrard-Bishop-Dauxois (PBD) model of DNA denaturation, although successful in the description of melting profiles, fails to predict melting entropies, unzipping forces and dynamical properties, e.g. hairpin dynamics. The paper presents an atomistic "toy model" of the intra base-pair motion which suggests that the thermodynamics may be better described by a planar vector - rather than a scalar - order parameter. This leads to correct estimates of melting entropy, unzipping force, hairpin opening rates, and the equilibrium constant of open/closed base pair states during imino proton exchange.

Pixelated plastic scintillator arrays can serve as high efficiency and high resolution neutron imaging detectors. Manufacturing these arrays is intensive in both time and labor. This work presents a fabrication method based on additive manufacturing for two-dimensional plastic organic scintillator arrays using a custom-built automated assembly machine and a custom photocurable resin that has significant non-aromatic acrylate oligomer content. The process involves two main stages: fully autonomous production of one-dimensional layered arrays, followed by semi-autonomous cutting and stacking to form two-dimensional pixel arrays. One-dimensional arrays were manufactured at a rate of around 4 layers per hour with minimal defects and tight dimensional tolerances, while two-dimensional arrays up to 7 x 7 pixels and 70 mm in length were completed in approximately 3.5 hours. Final arrays exhibited dimensional deviations of less than 0.5 mm. Two-dimensional arrays read out by a multi-anode photomultiplier tube demonstrated per-pixel position resolution and pulse-shape discrimination, enabling gamma-neutron interaction separation in mixed radiation environments.

We consider the stability of periodic gravity-capillary waves of finite amplitude for small values of the surface tension. Linear stability with respect to both superharmonic and subharmonic perturbations is calculated for each solution, and our methodology obtains the full eigenvalue spectrum consisting of growth rates and temporal frequencies. For small surface tension, the gravity-capillary wave solution space consists of a countably-infinite number of solution branches that coalesce in the small-surface-tension limit, which forms one of the main complications of our study. When the energy is fixed as an amplitude constraint, we find that the superharmonic instability associated with near-limiting gravity waves emerges at smaller amplitudes in the presence of surface tension. Further, the modulational (long-wave) instability is seen to be stabilised for finite-amplitude solutions in the presence of surface tension. This occurs at surface tension values well below that previously obtained via weakly-nonlinear theory, and the stabilisation is nonmonotonic as very small fluctuations in the surface tension of solutions produce large changes in their stability properties.

In the weakly collisional quiet-Sun corona, the electron distribution departs from Maxwellian, and “the electron temperature” is not single-valued: two standard diagnostics differ by a stable factor $R \approx 2.4$ across the solar cycle. We show this departure is a single relative entropy whose two orthogonal components are the EUV–radio diagnostic disagreement and the failure of the local conductive closure. By the generalized Pythagorean identity for Bregman divergences, the relative entropy of the $\kappa \approx 2.5$ distribution from its radio-diagnostic Maxwellian projection ($1.20$ nats) partitions with no cross-term into two components. The first is the energy-matched closure deficit ($0.32$ nats), the information discarded by the local Spitzer–Härm closure (Edmonds 2026c), which itself has no convergent form across $\kappa \in [2,3]$. The second is the inter-projection diagnostic gap ($0.876$ nats), what the EUV and radio diagnostics disagree by and a closed-form function of $R$ within the shape envelope. Closure failure and diagnostic disagreement are therefore the same relative entropy in two orthogonal coordinates: together they account for the corona's full departure from Maxwellian. The physical heat flux that survives is non-local and tail-dominated. Any mechanism that sets this $\kappa$ must act on the suprathermal tail, operate non-locally, and fix $\kappa$ to the observed band; of the standalone candidates only velocity filtration (Scudder 1992) meets the first two, and whether it, or anything, fixes $\kappa$ is the open problem we leave posed.

Liquid films on moving substrates are used in dip-coating processes to form uniform protective layers. Controlling free-surface waves is essential due to the film's inherent linear instability. Therefore, we develop a linear feedback controller to regulate the film toward a desired flat state by modulating the free-surface shear and pressure, with feedback gains derived analytically from linearised equations. Control performance is assessed for finite-amplitude waves using a Weighted Integral Boundary-Layer (WIBL) model at reduced Reynolds number $\delta = 8$. We identify parameter regimes in which pressure feedback is linearly destabilising while shear is stabilising, and vice versa, with the control mechanisms determined by the balance between the kinematic and dynamic wave velocities. Both stabilising and destabilising combinations of feedback coefficients can drive finite-amplitude waves toward the flat state $\bar{h}=1.1$ in finite time. In pressure-unstable regimes, the control induces a limit-cycle behaviour, in which long waves decay slowly due to the interplay between thickness and slope terms. The travelling-wave solution, although it decays slowly, moves against gravity, whereas other combinations reduce the wave amplitude in the direction of uncontrolled propagation. These results provide a foundation for higher-Reynolds-number studies and the design of industrially feasible actuator layouts.

Current global precipitation estimates from spaceborne precipitation radars are limited by their sensitivity to light and frozen precipitation, leading to systematic underestimation of precipitation at high latitudes. Because passive microwave retrievals (PMW) are commonly trained using these radar observations as reference data, this limitation is propagated into PMW This study introduces a novel PMW oceanic precipitation retrieval, GPROF-NN eXtended Precipitation Regime (XPR), that combines reference estimates from a cloud radar and a precipitation radar to overcome the sensitivity limitations of current spaceborne precipitation radars. The retrieval is trained to estimate light precipitation from CloudSat observations and moderate-to-heavy precipitation using observations from the GPM Dual-Frequency Precipitation Radar. The two estimates are combined using a fusion scheme to obtain a consistent precipitation estimate across precipitation regimes. Validation against in situ measurements from shipborne disdrometers shows a 26% improvement in the detection skill for high-latitude precipitation in terms of the critical success index and a reduction in the underestimation of high-latitude and frozen precipitation by more than 50% compared to retrievals constrained only by precipitation radar data. However, the fused retrieval does not improve the precision of instantaneous precipitation estimates, which is likely due to significant random errors in the CloudSat-based reference estimates of liquid precipitation. These results demonstrate that PMW retrievals can leverage the complementary sensitivities of cloud and precipitation radars to provide more consistent precipitation estimates across precipitation regimes than either reference instrument alone. The proposed retrieval provides a pathway to improve the representation of oceanic precipitation in future GPM precipitation products.

This study derives a scaling law connecting the Nusselt (Nu) and Bejan (Be) numbers in natural convection. Combining entropy generation analysis with boundary-layer scaling, the relation Be^-1 - 1 = a Nu^b naturally emerges without explicit dependence on geometry or boundary conditions. This is achieved within the present scaling framework when transport is governed by a single control parameter. Numerical validation against several cases corroborates this scaling. This finding reveals a direct, quantitative link between heat transfer efficiency and thermodynamic irreversibility, suggesting a potentially universal constraint that governs convective transport.

Vibrational wavepacket dynamics in the ground (X) and excited (B) electronic states of iodine under impulsive-pump/broadband-probe excitation are revisited. A method for accurate chirp correction, necessary to determine the zero time for each component of spectrally dispersed data and thereby separate coherent vibrational dynamics from coherent artifacts and population kinetics, is introduced. While from these processed time-domain data the absolute Raman cross-section in the ground electronic state can be calculated using steady-state absorption, we show that the same can be done using the pump-probe data itself, and further extend this method as a benchmark to calculate the same for the excited electronic state; these cross-sections report on vibronic couplings specific to these states. Further, since the Fourier transform of the processed data yields information on vibrational modes averaged over the dephasing time, a wavelet analysis is performed to yield a joint time-frequency distribution of the vibrational modes, demonstrating how the time evolution of their frequencies can be extracted. The vibrational modes of the ground and excited electronic states are shown to exhibit distinct dispersion characteristics. Since overlapping spectral features appear at different time windows, such an analysis can disentangle spectral congestion, even from a simple one-dimensional measurement. Most interestingly, a rapid time-dependent spectral shift and decay of the B state mode, followed by the appearance and growth of the A-state mode, directly correlates with the pre-dissociation, followed by solvent caging-induced recombination. Thus, the present work reveals transfer of vibrational coherence from one electronic state (B) to another (A), mediated via nonadiabatic coupling to the intermediate dissociative state (a), underscoring the importance of electronic coherence.

Building on the phase reduction theory formulated for reaction-diffusion systems with spatial translational symmetry, we develop a data-driven method that reconstructs the spatiotemporal phase dynamics of traveling and oscillating patterns. Spatiotemporal phase dynamics are described by spatial and temporal phases that represent the position and oscillation of the pattern, respectively. Using Bayesian inference, our method directly reconstructs phase equations from time-series data. When tested on simulation data from coupled Gray-Scott models exhibiting traveling breathers, the method accurately reconstructs the deterministic part of the phase equations in the weak-noise regime, in which the phase dynamics converge to a linearly stable fixed point.

We formulate a thermodynamically consistent continuum theory for compressible, viscous, heat-conducting fluids in which the velocity entering the balance of mass is distinguished from the specific linear momentum entering the balances of linear momentum and energy. Starting from balances of mass, linear momentum, angular momentum, and internal energy, together with a power identity and the Clausius–Duhem inequality, we derive the mechanical and thermodynamic consequences of allowing these fields to differ. From local angular-momentum balance, we show that the Cauchy stress need not be symmetric and we determine its skew part. From the dissipation inequality, we obtain an admissible internal-energy flux and a closure in which the relative transport between mass and linear momentum is proportional to the pressure gradient rather than to the mass-density gradient. We also derive a free-enthalpy imbalance across shocks and a reduced wall dissipation inequality for rigid, impermeable walls undergoing prescribed rigid motion, together with simple admissible wall laws for temperature-controlled and heat-flow-controlled settings. For ideal gases, we write the governing equations in conservative dimensionless form, recover the classical compressible Navier–Stokes–Fourier theory when relative transport vanishes, and identify a distinguished low-Mach regime in which mass transport and linear-momentum transport remain distinct at leading order.

Principled prediction of when and where links form in complex networks is a fundamental problem. We derive a closed-form non-Markovian expression for next-step connection probabilities that unifies latent hyperbolic geometry with long-range memory of past interactions. This expression yields interpretable forecasts governed by a small set of parameters. Applied to large-scale real networks, we find quantitative agreement with empirical connection probabilities and reveal how geometry and memory jointly shape link dynamics. These results establish a minimal and extensible foundation for principled probabilistic forecasting of temporal network topology.

Computational fluid dynamics (CFD) simulations play an important role in engineering science and applications, however, it is not applicable for problems requiring a large number of repeated calculations. Accordingly, many reduced-order modelling techniques are developed to reduce computational costs, improve the efficiency, and have achieved significant progress. At present, most studies focus on reconstructing the flow field throughout the parameter space of the snapshots within a fixed time window. However, the prediction problem has always been challenging, especially for unsteady flow. In this work, a reduced-order model (ROM) based on proper orthogonal decomposition (POD) and radial basis function (RBF) is presented and applied to the prediction problem of an unsteady flow with periodic changing boundary conditions. The method is validated by a numerical case of three-dimensional unsteady flow around cylinders with time-varying inlet velocity. This method is demonstrated to be quite accurate and efficient, reducing the CPU time by more than 99% with an accuracy loss less than 5.2% for predictions.

Variable flux memory motors, which employ Low Coercive Force (LCF) magnets, achieve extended high-efficiency operation through controllable magnetization states. To address the need for a unified approach to defining and comparing the magnetization state (MS) across material and motor levels, this paper proposes four MS definitions: two based on intrinsic material properties-magnetic flux density B and magnetic polarization J-and two based on motor-level quantities-fundamental flux linkage and back-EMF components. These definitions are evaluated across the id, iq operating plane using finite element analysis on an interior PMSM with a hybrid magnet configuration (LCF and HCF: High Coercive Force) and a defined circuit setup. The results clarify the relationship between material-level behavior and measurable motor quantities. The proposed framework provides guidance for selecting appropriate MS metrics depending on the application objective, whether for material analysis, control implementation, or condition monitoring in variable flux machines.

Since its first successful applications in the early 2000s, x-ray Thomson scattering (XRTS) has emerged as one of the most successful tools for the diagnostics of extreme states of matter in the laboratory. By sampling the dynamic structure factor of the electrons, XRTS is capable of giving detailed insights into the atomic-scale physics of the matter probed. Moreover, thermodynamic parameters, like the mass density, temperature, and ionization state, are routinely inferred from XRTS measurements, providing a comprehensive characterization of the sample probed. In addition, the dynamic structure factor is of considerable interest in its own right as it contains information on other effects such as the plasmon shift, miscibility between species, electronic states and potential transitions between these states. In this work, we provide an extensive overview of previous XRTS experiments at both traditional laser and X-ray free electron laser facilities, including information about the probed material (elements, conditions), scattering geometry, analysis methods as well as corresponding references. In addition, we briefly discuss the advantages and shortcomings of widely used analysis methods for XRTS spectra and reflect on upcoming future developments in XRTS experiments and theory.

This work presents GS-DOT, a novel image reconstruction framework based on Gaussian Splatting (GS) for diffuse optical tomography (DOT). Inspired by GS for rendering applications, absorption coefficients are represented as a sparse sum of anisotropic Gaussian primitives optimized to fit measured time-resolved point-spread functions through analytic gradients and Adam optimization. This is the first adaptation of GS algorithms in the photon diffusion regime, where the ray transport function is replaced by the diffusion functions to enable accurate modeling of light transport in highly scattering media. Validation on synthetic tissue models demonstrate high accuracy in localization and quantification of reconstructed absorption maps for both clean and noisy signals. GS-DOT has demonstrated high robustness to noise and showed a huge reduction in memory demand.

Recent developments have seen the application of finite Gaussian basis sets to the $\alpha(Z\alpha)^{n\geq3}$ vacuum polarization. The energy shift for $s$ and $p$ electron states have been tabulated and their convergence investigated. In this work, we extend this problem to the multi-electron case. Hartee-Fock potentials obtained self-consistently are used to treat the vacuum polarization for lithium-like systems and are found to be in good agreement with comparable results in the literature. The results presented in this work demonstrate the use of Gaussian basis sets for atomic potentials whose Green's functions expressions cannot be simply obtained via analytic or numerical methods.

We study the influence of capillary effects on the motion of thin elastic plates denser than water drifting in propagating surface gravity waves. Such floaters experience a mean angular drift that rotates them toward two preferential orientations: parallel to the direction of wave propagation (longitudinal) or parallel to the wave crests (transverse). We develop a diffractionless model (Froude-Krylov approximation) to compute the mean yaw moment acting on floaters with arbitrary bending rigidity, small relative to the wavelength. Capillary forces are incorporated through a quasi-static volume formulation based on the fluid volume displaced by the floater and its meniscus. The model predicts that the preferential orientation is governed by the non-dimensional parameter $F = kL_x^2/\overline{h}$ recently introduced in Herreman et al. (J. Fluid Mech., vol.999, 2024, A92), where $k$ is the wavenumber, $L_x$ the floater length, and $\overline{h}$ the equilibrium immersion depth, provided that $\overline{h}$ accounts for capillary effects. The orientation depends on how $F$ compares to a critical value $F_c$, which is a function of the ratio of the flexural length to the floater length. These predictions are in good agreement with experiments performed with thin metal rectangular plates of various length, width and thickness.

Spectral problems governed by differential operators underpin a wide range of physical systems, yet remain computationally challenging because their spectra depend sensitively on continuous parameters and often demand repeated evaluations across parameter space. Here we present $\texttt{DeepOPiraKAN}$, an open source physics informed neural network architecture for spectral analysis. By combining operator learning with enhanced optimization stability, it captures the underlying parameter-to-spectrum mapping in a single model, avoiding repeated spectral solutions at isolated points in parameter space. As a representative and stringent benchmark, we apply this framework to the computation of quasinormal modes of Kerr black holes. A single trained network accurately resolves modes with $(\ell,m)\in \{(2,0),(2,1)\}$ and overtones up to $n=7$ across the full spin range, achieving relative errors of $\mathcal{O}(10^{-6})$ for the fundamental mode and gradually increasing to $\mathcal{O}(10^{-4})$ for higher overtones, benchmarked against the Leaver's method. This level of accuracy is already significant for black hole spectroscopy and practical ringdown modelling for current and future observatories. More broadly, these results highlight the potential of $\texttt{DeepOPiraKAN}$ as a general and scalable framework for parameter dependent spectral problems across complex physical systems.

A microbubble is, in essence, a fully enclosed thin-walled microcavity. Unlike spherical microbubbles formed by expansions, 3D printing enables the free definition of their geometry, allowing precise control over shape and dimensions during fabrication. However, the geometric nature of microbubbles poses significant challenges for conventional photoresist-based lithographic microfabrication due to their fragile thin-walls, enclosed hollow volumes, and high sensitivity to mechanical stresses. These characteristics prevent developer solvents from accessing the internal cavities to remove unexposed photoresist. Two-photon polymerisation (2PP) is a laser-based 3D microprinting technique capable of sub-diffraction-limited resolution, offering exceptional design freedom for fabricating complex micro-architectures in photoresists. In this study, we demonstrate a 2PP-based method that overcomes these limitations and, for the first time, enables the successful fabrication of cubic microbubbles with ultra-high-aspect-ratio thin walls and fully enclosed microcavities using high-viscosity SU-8 2050 photoresist. The optimised process parameters and structural design facilitate efficient extraction of unexposed photoresist from the cavity interior while achieving a thin-wall ultra-high aspect ratio of approximately 340:1. The hollow nature and mechanical integrity of the printed structures were experimentally confirmed using micromanipulator-based probing. The proposed method maintains excellent dimensional accuracy and significantly reduces printing time for large-scale and high-count builds in 2PP processes. Such microbubbles are fundamental building blocks for optical resonators, microelectromechanical systems (MEMS) pressure sensors, microfluidic reaction chambers, and emerging metamaterials.

Natural disaster strikes at any given moment from seemingly out of nowhere Akin to earthquake that strongly affects human with different magnitudes through the course of time. The main aim of this study is the fractal analysis of seismic activity data of India in the interval from 04-10-2016 to 31-05-2023. This includes analyzing the earthquake magnitudes and their epicenters using fractal statistics, which were studied at different scales to identify patterns in the data through the use of the fractal spectrum. The probabilities of future earthquakes with different magnitudes were estimated using the fractal model.

The Harappan Civilization (c.\ 2600–1900 BCE) achieved extraordinary standardization across a vast region without clear evidence of centralized rule. We propose that it was a merchant commonwealth organized as a scale-free commercial network, with AP2PBS (Asynchronous Peer-to-Peer Barter Settlement) and the Unicorn Logistics Cartel (ULC) serving as inferential model labels for its coordinating institutions. Four lines of evidence support this interpretation. (1) Statistical: offering stand frequencies follow a power-law distribution with $\alpha \approx 2.3$–$2.6$, while seal counts across sites follow $\alpha \approx 1.55$; both fit heavy-tailed commercial organization better than exponential alternatives. (2) Attribute correlations: non-random associations between offering stands and anatomical attributes, including Cramér's V patterns across market tiers, are consistent with deliberate institutional design of visual confirmation bundles. (3) Archaeological: seal evolution, contextual distributions, and warehouse logistics support the operational requirements of long-distance asynchronous exchange. (4) Architectural: Mohenjo-daro's halls, bath, warehouses, and cart-compatible gridiron are consistent with infrastructure for commerce and coordination. Taken together, these findings are consistent with AP2PBS and ULC as models of how independent participants could standardize authentication, exchange value asynchronously, and coordinate across distance without a hierarchical state. Scale-free topology also offers a testable explanation for both Harappan longevity and its vulnerability to hub failure and cascading decline. The argument is inferential: the archaeological record does not directly identify AP2PBS or ULC by name, but its statistical, material, and architectural patterns are compatible with this decentralized commercial model.

Topological quasiparticles of light, such as classical and quantum optical skyrmions, have so far relied on fully coherent or pure quantum states whose topology is encoded in the entanglement between polarization and two-dimensional spatial modes. Here we show that skyrmionic topology can emerge directly within the density matrix of a mixed quantum state. We introduce a framework in which a coherence-Stokes vector defines a topological texture over the density matrix, enabling the realization of quantum skyrmions using only a pseudospin and a real or synthetic one-dimensional space of modes. For a single photon, the density matrix is analogous to the coherence matrix of classical light, and can be encoded using partially coherent electromagnetic fields. We further analyze the topological texture encoded in a bipartite entangled photon pair, showing how skyrmions arise not only in the full bi-photon system, but also in its reduced subspaces with any pseudospin-mode combination simultaneously. We then explore the robustness of such skyrmions to environmental noise, and discover their persistence in mixed-quantum states of multiple photons. Finally, we propose a feasible experimental route to generate and measure such skyrmions using integrated photonic networks, and suggest avenues for similar implementations in other quantum systems. Our work paves the way for robustly encoding classical information on partially coherent light or on mixed quantum states and for encoding topological charges on many-body quantum systems.

Controlling and generating ultrafast free-electron wavepackets via laser is pivotal for photon-induced near-field electron microscopes (PINEM) and also for developing compact, coherent free-electron radiation sources. Here, we present a quantum echo-enabled high-harmonic generation (QEEHG) scheme that manipulates the quantum phase of electron wavepackets to produce tunable, coherent high-harmonic radiation at ultrashort wavelengths. This framework, inspired by the EEHG concept for free-electron lasers by Stupikov et al. (2009), leverages multiphoton PINEM scattering followed by dispersive chirp sections to induce quantum interference among photon sidebands. Such interference selectively enhances a targeted harmonic order - for instance, the 60th harmonic at 13.3nm from an 800nm seeding - while suppressing unwanted radiations. The optimization of harmonic orders and its non-classical spectral characteristics are analyzed. This quantum echo technique establishes a promising paradigm for compact coherent sources and provides new perspectives for quantum wavefunction shaping in ultrafast electron microscopy and diffraction.