We present a novel self-consistent theoretical framework to characterize the formation, evolution, and merger sites of dynamically-formed black hole binaries, with a focus on explaining the most massive events observed by the LIGO-Virgo-KAGRA Collaboration. Our approach couples the galaxy formation model GAMESH with cluster population synthesis codes to trace the cosmic evolution of globular clusters simultaneously with mergers of massive black holes. Our reference model, which includes prescriptions for both cluster formation and disruption depending on properties of specific galaxies, accurately reproduces the observed age-mass distribution of the Milky Way globular clusters. We find that approximately 30% of the globular clusters observed in our galaxy's halo may have originated from satellite galaxies of the Milky Way. We confirm that hierarchical black hole mergers provide a significant contribution to the formation of black holes in and above the pair-instability mass gap. However, quantifying their contribution is challenging, as different population synthesis codes yield divergent results in terms of black hole mass function and merger rates. Furthermore, we characterize the host galaxies where massive black holes form in terms of their dark matter, stellar mass, and metallicity. Ultimately, we demonstrate that the merger and birth rate densities of binary black holes increase with redshift till z = 5. This cosmic evolution is a crucial signature with significant implications for future detectors like the LISA, the Einstein Telescope and Cosmic Explorer, which will be capable to probe the high-redshift Universe.

The least absolute shrinkage and selection operator (Lasso) is a popular method for high-dimensional statistics. However, it is known that the Lasso often has estimation bias and prediction error. To address such disadvantages, many alternatives and refitting strategies have been proposed and studied. This work introduces a novel Lasso--Ridge method. Our analysis indicates that the proposed estimator achieves improved prediction performance in a range of settings, including cases where the Lasso is tuned at its theoretical optimal rate \(\sqrt{\log(p)/n}\). Moreover, the proposed method retains several key advantages of the Lasso, such as prediction consistency and reliable variable selection under mild conditions. Through extensive simulations, we further demonstrate that our estimator outperforms the Lasso in both prediction and estimation accuracy, highlighting its potential as a powerful tool for high-dimensional linear regression.

Effects of General Relativity are usually neglected in the non-linear evolution of structures, where Newtonian $N$-body simulations are traditionally employed. In the post-Friedmann expansion framework, a weak-field relativistic approximation purpose-built for cosmology, a frame-dragging gravito-magnetic vector potential arises at leading order, sourced by momentum currents. At this order, the vector potential contributes to the metric while leaving the dynamics of the matter fields unaffected, as it does not appear in the Euler equation. It can therefore be extracted a posteriori from standard N-body simulations, where the dynamics is purely Newtonian. Using the Delaunay Tessellation Field Estimator code on the IllustrisTNG simulations, here we extend previous work in order to compute the power spectrum of this vector potential down to galactic scales. The magnitude of the vector potential is two orders of magnitude larger than predicted by perturbation theory, and is a $1\% \sim 0.1\%$ effect compared to the non-linear Newtonian scalar gravitational potential. In the redshift range considered here, the gravito-magnetic effect remains subdominant, without showing any enhancement during a particular phase in the evolution of structures, aside from the continuous growth of non-linearity at low redshift. Although this seems to suggest that, within the $Λ$CDM model, no significant gravito-magnetic effects contribute to the non-linear evolution of cosmic structures, i.e. to the dynamics of massive particles, possible observational consequences, e.g. in lensing, deserve further exploration.

Using public pulse time-of-arrival data from five pulsar timing arrays (PTAs), we search for a stationary, isotropic, and unpolarized nHz stochastic gravitational-wave background (SGWB). This analysis is more sensitive than previous individual PTA searches because the combined 121-pulsar dataset is about four times larger than any single PTA's. For pulsars observed by multiple PTAs, we employ a new "direct combination" method to merge their astrophysical models and data. This avoids the challenge of reconciling different PTA timing models to obtain a single "best" model. A central result of our analysis is the posterior distribution of the amplitude $A_{gw}$ and exponent $γ_{gw}$ of the putative SGWB energy-density spectrum, modeled as a power law in frequency. While these results are consistent with a nonzero SGWB amplitude $A_{gw}$, the statistical significance-assessed via a Bayesian odds ratio and noise-marginalized false-alarm probabilities ($p$-values) for three detection statistics-remains below the conventional $5σ$ threshold for a confident detection. The inter-pulsar timing-residual correlation, reconstructed as a function of angle $θ$ between the pulsar lines of sight, matches the Hellings and Downs (HD) prediction.

The solar wind is observed to undergo substantial heating as it expands through the heliosphere, with measured temperature profiles exceeding those expected from adiabatic cooling. A plausible source of this heating is reflection-driven turbulence (RDT), in which gradients in the background Alfvén speed partially reflect outward-propagating Alfvén waves, seeding counter-propagating fluctuations that interact and dissipate via turbulence. Previous RDT models assume a radial background magnetic field, but at larger radii the interplanetary field is known to be twisted into the Parker Spiral (PS). Here, we generalize RDT phenomenology to include a PS, using three-dimensional expanding-box magnetohydrodynamic (MHD) simulations to test the ideas and compare the resulting turbulence to the radial-background-field case. We argue that the underlying RDT dynamics remain broadly similar with a PS, but the controlling scales change: as the azimuthal field grows it "cuts across" perpendicularly stretched, pancake-like eddies, producing outer scales perpendicular to the magnetic field that are much smaller than in the radial-background case. Consequently, the outer-scale nonlinear turnover time increases more slowly with heliocentric distance in PS geometry, weakening the tendency (seen in radial-background models) for the cascade to 'freeze' into quasi-static, magnetically dominated structures. This allows the system to dissipate a larger fraction of the fluctuation energy as heat, also implying that the turbulence remains strongly imbalanced (with high normalized cross-helicity) out to larger heliocentric distances. We complement our heating results with a detailed characterization of the turbulence (e.g., spectra, switchbacks, and compressive fractions) providing a set of concrete predictions for comparison with spacecraft observations.

We present a classroom based activity using a sunrise photograph of Mont Blanc shadow taken from Geneva to estimate the Earth radius. By determining the direction of solar rays relative to the local vertical and accounting for atmospheric refraction, students can derive an upper bound approximately 70 percent above the accepted value. The discrepancy provides a concrete illustration of modelling assumptions and observational limitations. The activity combines geometric reasoning, basic trigonometry, and order of magnitude estimation, allowing students to obtain a physically meaningful result from simple observations. Beyond its quantitative aspects, the approach highlights essential elements of the scientific method, including hypothesis formulation, model construction, uncertainty analysis, and comparison with external data, offering a structured introduction to Nature of Science concepts within upper secondary STEM curricula.

This work presents the discontinuous Galerkin discretization of the consistent splitting scheme proposed by Liu [J. Liu, J. Comp. Phys., 228(19), 2009]. The method enforces the divergence-free constraint implicitly, removing velocity--pressure compatibility conditions and eliminating pressure boundary layers. Consistent boundary conditions are imposed, also for settings with open and traction boundaries. Hence, accuracy in time is no longer limited by a splitting error. The symmetric interior penalty Galerkin method is used for second spatial derivatives. The convective term is treated in a semi-implicit manner, which relaxes the CFL restriction of explicit schemes while avoiding the need to solve nonlinear systems required by fully implicit formulations. For improved mass conservation, Leray projection is combined with divergence and normal continuity penalty terms. By selecting appropriate fluxes for both the divergence of the velocity field and the divergence of the convective operator, the consistent pressure boundary condition can be shown to reduce to contributions arising solely from the acceleration and the viscous term for the $L^2$ discretization. Per time step, the decoupled nature of the scheme with respect to the velocity and pressure fields leads to a single pressure Poisson equation followed by a single vector-valued convection-diffusion-reaction equation. We verify optimal convergence rates of the method in both space and time and demonstrate compatibility with higher-order time integration schemes. A series of numerical experiments, including the two-dimensional flow around a cylinder benchmark and the three-dimensional Taylor--Green vortex problem, verify the applicability to practically relevant flow problems.

This study investigates the signatures of pressure extrema on global oscillations in discs. To this end, we use the framework of wave topology to establish a generalised local dispersion relation that includes pressure gradients. We highlight the influence of a previously unrecognized epicyclic-acoustic frequency and derive an analytical criterion for the existence of a branch of modes transiting between the inertial and the pressure bands. We find that pressure extrema consist of wave guides in which such topological modes propagate. The fundamental mode trapped at a pressure bump can propagate at all frequencies, allowing it to resonate with any temporal forcing, while the mode associated with a pressure gap propagates at a fixed frequency, propagates with arbitrary vertical phase velocity. These specific features make them attractive candidates for future discoseismology.

This paper presents a method for solving the Inverse Stochastic Differential Game (ISDG) problem in finite-horizon linear-quadratic Gaussian (LQG) differential games. The objective is to recover cost function parameters of all players, as well as noise scaling parameters of the stochastic system, consistent with observed trajectories. The proposed framework combines (i) estimation of the feedback strategies, (ii) identification of the cost function parameters via a novel reformulation of the coupled Riccati differential equations, and (iii) maximum likelihood estimation of the noise scaling parameters. Simulation results demonstrate that the approach recovers parameters, yielding trajectories that closely match the observed trajectories.

The continuous spectrum of double beta decay ($ββ$) provides a sensitive probe to test the predictions of the standard model and to search for signatures of new physics beyond it. We present a comprehensive analysis of the $^{136}$Xe $ββ$ spectrum utilizing $39.1 \pm 0.7~\textrm{kg}\cdot\textrm{yr}$ of $^{136}$Xe exposure from the PandaX-4T experiment. The analysis yields the most precise measurement to date of the $^{136}$Xe two-neutrino double beta decay ($2νββ$) half-life, $(2.14 \pm 0.05) \times 10^{21}$ years, the uncertainty of which is reduced by a factor of 2 compared to our previous result. We measure the parameter $ξ_{31}^{2ν}$, defined as the ratio between the subleading and leading components of the $^{136}$Xe $2νββ$ nuclear matrix element, to be $0.59^{+0.41}_{-0.38}$, which is consistent with theoretical predictions. We also search for Majoron-emitting modes of $^{136}$Xe $ββ$, establishing the most stringent limit for the spectral index $n=7$.

The prospect of merging the paradigms of geometric frustration on a triangular lattice and bond anisotropies in the strong spin-orbit coupling limit holds tremendous promise in the ongoing hunt for exotic quantum materials. Here we identify a new candidate system to realize such physics, the organic quantum antiferromagnet (CD$_3$ND$_3$)$_2$NaRuCl$_6$. We report a combination of thermodynamic, magneto-elastic and neutron scattering experiments on single-crystals to determine the phase diagram in axial magnetic fields $\mathbf{H \parallel c}$ and propose a minimal model Hamiltonian. (CD$_3$ND$_3$)$_2$NaRuCl$_6$ displays an ideal triangular arrangement of Ru$^{3+}$ ions adopting the spin-orbital entangled $j_{\rm eff} = 1/2$ state. It hosts residual magnetic order below $T_{\rm N} = 0.23$ K and a highly unusual $H-T$ phase diagram including three different incommensurate states. Spin-waves in the high-field polarized regime are well described by a Heisenberg-like triangular lattice Hamiltonian with a potential sub-leading bond dependent anisotropy term $J_{\pm\pm}$. We discuss possible candidate magnetic structures in the various observed phases and propose two mechanisms that could explain the field-dependent incommensurability, requiring either a small ferromagnetic Kitaev term or a tiny magneto-elastic $J-J'$ isosceles distortion driven by pseudospin-lattice coupling. We argue that the multi-$\mathbf{q}$ ground state in zero magnetic field is a prime candidate for hosting the $\mathbb{Z}_2$ vortex crystal proposed on the triangular Heisenberg-Kitaev model. (CD$_3$ND$_3$)$_2$NaRuCl$_6$ is the first member in an extended family of quantum triangular lattice magnets, providing a new playground to study the interplay of geometric frustration and spin-orbit effects.

Primordial Magnetic Fields (PMFs) -- magnetic fields originating in the early Universe and permeating the cosmological scales today -- can explain the observed microGauss-level magnetisation of galaxies and their clusters. In light of current and upcoming all-sky radio surveys, PMFs have drawn attention not only as major candidates for explaining the large-scale magnetisation of the Universe, but also as potential probes of early-Universe physics. In this paper, using cosmological simulations coupled with light-cone analysis, we study for the first time the imprints of the PMF structure on the mean rotation measure (RM) originating in the intergalactic medium (IGM), $\langle \mathrm{RM_{IGM}}\rangle$. We introduce a new method for producing full-sky $\mathrm{RM_{IGM}}$ distributions and analyse the autocorrelation of $\mathrm{RM_{IGM}}$ on small and large angular scales; we find that PMF structures indeed show distinct signatures. The large-scale uniform model (characterised by an initially unlimited coherence scale) leads to correlations up to 90 degrees, while correlations for small-scale stochastic PMF models drop by factor of $100$ at $ 0.17, 0.13$ and 0.11 degrees angular scales, corresponding to $5.24, 4.03$ and $3.52$ Mpc scales (at $z=2$ redshift) for magnetic fields with comoving $3.49, 1.81, 1.00 $ Mpc/h coherence scales, respectively; the correlation amplitude of the PMF model with comoving $\sim 19$ Mpc/h coherence scale drops only by factor of $10$ at 1 degree (30.6 Mpc). These results suggests that improvements in the modelling of Galactic RM will be necessary to investigate the signature of large-scale correlated PMFs. A comparison of $\langle \mathrm{RM_{IGM}}\rangle$ redshift dependence obtained from our simulations with that from the LOFAR Two-metre Sky Survey shows agreement with our previous upper limits' estimates on the PMF strength derived from RM-rms analysis.

We present a comprehensive study of the magnetocaloric effect (MCE) in the Nd$T_4$B system where $T$ = Fe, Co, and Ni. These compounds are ferromagnetic kagome materials with tunable ordering temperatures, transition width, and magnetic moments depending on the choice of transition metal. Thus, they are good candidates for investigating the MCE. We characterize the MCE using standard metrics and construct ternary phase diagrams as functions of Fe, Co, and Ni concentrations. Using these phase diagrams, we engineer the composition NdFe$_{1.15}$Co$_{0.46}$Ni$_{2.39}$B to maximize the MCE. Interestingly, the Nd$T_4$B system shows a notable entropy change over a wide temperature range ($\sim$10 to 650 K), and particular compositions have notable MCEs spanning hundreds of Kelvin, making this a suitable system to study for technologies used in a wide range of temperatures. In a few cases, we observe a two-peak MCE. These two transitions, releasing comparable entropy, provide an interesting platform to study for applications in multi-stage cooling.

Passively cooling neutron stars (NSs) should reach undetectably low surface temperatures $T_s<10^4$ K in less than $10^7$ yr. However, HST observations have revealed likely thermal UV emission from the Gyr-old millisecond pulsars PSR~J0437$-$4715 and PSR~J2124$-$3358, and from the $\sim10^{7-8}$ yr-old classical pulsars PSR~B0950$+$08 and PSR~J0108$-$1431, implying $T_s\sim10^5$ K and the need for heating mechanisms. We compute the thermal evolution of these NSs including rotochemical heating (RH) in the core with normal or Cooper-paired matter, vortex creep (VC) in the inner crust, and crustal heating through nuclear reactions, and compare the results with observations and with the upper limit for PSR~2144$-$3933. No single mechanism explains all sources. The high temperature of PSR~J0437$-$4715 can be reproduced by RH with a large Cooper pairing gap $Δ_i\sim1.5$ MeV for either neutrons or protons, but this requires an unrealistically short initial period $P_0\lesssim1.8$ ms to activate the same mechanism in PSR~B0950$+$08. Conversely, the latter can be explained by RH with modified Urca reactions in normal matter or by VC with an excess angular momentum $J\sim3\times10^{43}$ erg,s, but these models underpredict PSR~J0437$-$4715. A model combining RH with a large pairing gap and VC matches both pulsars and is consistent with the upper limits for the remaining three. It further predicts that their temperatures should lie close to these limits, suggesting that deeper or broader-wavelength observations would provide a strong test of this scenario.

We propose high-fidelity single-qubit spin-dependent kicks (SDKs) for trapped ions using nanosecond Raman pulses via amplitude modulation of a continuous-wave laser with a tunable beat frequency. We develop a general method for maintaining SDK performance in the presence of micromotion by identifying optimal choices of the RF phase and frequency that suppress unwanted backward kicks. The proposed scheme enables SDK infidelities as low as $10^{-9}$ in the absence of micromotion, and below $10^{-5}$ with micromotion. This study lays the foundation for the realization of sub-trap-period and high-fidelity two-qubit gates based on SDKs.

As measurements of neutrino mixing angles continue to become more precise, it is increasingly likely that in the very near future a realistic neutrino mixing model will be required to precisely reproduce their best-fit values. In this study, a modified TM$_2$ mixing model which reproduces the best-fit values of all three neutrino mixing angles is proposed. The model reproduces the correct mixing angles within 1$σ$ of the current best-fit values and is robust against any future changes of the best-fit values.

Understanding puffing topography (PT), which includes puff duration, intra-puff interval, and puff count per session, is critical for evaluating Electronic Nicotine Delivery Systems (ENDS) use, toxicant exposure, and informing regulatory decisions. We developed FRIENDS (Flexible Robust Instrumentation of ENDS), an open-source device that can be attached to ENDS and records puffing and touching events. This paper introduces the FRIENDS graphical user interface (GUI) that improves accessibility and interpretability of data collected by FRIENDS. The GUI is a Python-based opensource tool that extracts, decodes, and visualizes 24-hour puffing data from the FRIENDS device. Validation using 24-hour experimental data confirmed accurate timestamp conversion, reliable event decoding, and effective behavioral visualization. The software is freely available on GitHub for public use.

We use the extremely high-resolution ($m_{\rm bary}=4\rm{M}_\odot$) LYRA cosmological galaxy formation simulations of six dwarf galaxies with $M_{\rm 200c}\sim10^9\rm{M}_\odot$ at $z=0$ to investigate their stellar assembly histories. Based on the age of stars in these galaxies at $z=0$, $40-100\%$ of their stellar mass was formed by the time of reionization, when star formation (SF) abruptly shuts down. Depending on their halo mass evolution, some of the dwarfs reignite SF post-reionization (rejuvenators), while others remain quenched for the rest of cosmic time (reionization relics). However, the stellar mass of relics can still grow by more than $50\%$ through mergers post-reionisation. We find clear correlations between metallicity distributions of the galaxies and the fraction of stars formed post-reionization ($f_{\rm post-reio}^\star$) such that relics have lower median $\rm [Fe/H]$ with a more prominent low metallicity tail. Moreover, the shape of the galaxies at $z=0$ correlates with their $f_{\rm post-reio}^\star$, with rejuvenators showing more spherical stellar distribution than relics. This difference arises only post-reionization when rejuvenators become rounder with more SF activity. Similarly, the shape of dark matter (DM) halos in the inner regions display more spherical distributions in rejuvenators than in relics. The shape evolution shows that DM haloes in all galaxy formation simulations become rounder in comparison to their collision-less, DM-only counterparts. However, DM haloes of rejuvenators evolve more significantly. We do not find any correlation between SF activity and formation of shallow DM density cores in these galaxies. These predictions can be tested using upcoming observational data. In particular, our results indicate that the scatter in the mass-metallicity relation in the low mass regime is correlated with SF histories and the shape of galaxies.

The Hubble tension problem is one of the most significant challenges in modern cosmology. In this paper, we study the Hubble tension problem in the framework of holographic dark energy (HDE). To perform a systematic and comprehensive analysis, we select six representative theoretical models from all four categories of HDE. For the observational data, we adopt the Baryon Acoustic Oscillation (BAO) data from the Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2) along with a collection of alternative BAO measurements, Cosmic Microwave Background (CMB) distance priors from $Planck$ 2018, and type Ia supernovae (SN) data from the PantheonPlus, Union3, and DESY5 compilations. We find that HDE models that employ the Hubble scale or its combinations as the infrared (IR) cutoff cannot alleviate the Hubble tension problem. In contrast, HDE models that employ the future event horizon as the IR cutoff can partially mitigate the Hubble tension problem. It must be stressed that these two key conclusions hold true for cases of adopting different theoretical HDE models and different observational data. Our findings advocate for further exploration of HDE models using other types of cosmological observations.

Since Thurston pioneered the connection between circle packing (abbr. CP) and three-dimensional geometric topology, the characterization of CPs and hyperbolic polyhedra has become increasingly profound. Some milestones have been achieved, for example, Rodin-Sullivan \cite{Rodin-Sullivan} and Schramm \cite{schramm91} proved the rigidity of infinite CPs with the intersection angle $Θ=0$. Rivin-Hodgson \cite{RH93} fully characterized the existence and rigidity of compact convex polyhedra in $\mathbb{H}^3$. He \cite{He} proved the rigidity and uniformization theorem for infinite CPs with $0\leqΘ\leq π/2$. Therefore, the remaining unresolved issues are the rigidity and uniformization theorems for infinite CPs with $0\leqΘ<π$, as well as for infinite hyperbolic polyhedra. In fact, He specifically claimed in the abstract of \cite{He} that ``in a future paper, the techniques of this paper will be extended to the case when $0\leqΘ<π$. In particular, we will show a rigidity property for a class of infinite convex polyhedra in the 3-dimensional hyperbolic space". The objective of the article is to accomplish the work claimed in \cite{He} by proving the rigidity and uniformization theorem for infinite CPs with $0\leqΘ<π$, as well as infinite trivalent hyperbolic polyhedra. We will pay special attention to CPs whose contact graphs are disk triangulation graphs. Such CPs are called regular because they exclude some singular configurations and correspond well to hyperbolic polyhedra. We will establish the existence and rigidity of infinite regular CPs. Moreover, we will prove a uniformization theorem for regular CPs, which solves the classification problem for regular CPs. Thereby, the existence and rigidity of infinite convex trivalent polyhedra are obtained.

We systematically study how cooling creates structural changes in supernova remnants as they evolve. Inspired by the peculiar morphology of the Pa 30 nebula, we adopt a framework in which to characterize supernova remnants under different degrees of cooling. Our cooling framework characterizes remnants with a singular parameter called $β$ that sets how rapidly the system's thermal energy is radiated or emitted away. A continuum of morphologies is created by the implementation of different cooling timescales. For $β\gtrsim 400$, or when the cooling timescale is shorter than $\approx \frac{1}{400}$ of the Sedov time, the ejecta is shaped into a filamentary structure similar to Pa 30. We explain the filament creation by the formation of Rayleigh-Taylor Instability fingers where cooling has prevented the Kelvin-Helmholtz Instability from overturning and mixing out the tips. The ejecta in these filaments have not decelerated and are moving almost completely ballistically at $\approx 95-100\%$ their free expansion speed. In this rapid cooling regime, an explosion energy $\approx 3.5 \times 10^{47}$ erg is inferred. We also propose the cooling mechanism required to create these structures necessitates removing energy at a rate of $2\%$ of $E_{\rm ej}/t$, which implies a cooling luminosity of $\approx 10^{36}$ erg/s.

We analyzed TESS archival data of three novae after recent outbursts, searching the orbital and white dwarf (WD) rotation period and possible variations of these periods. In V1405 Cas, we detected a period of $\sim$116.88 seconds, which we identified as due to the WD spin, and measured a rate of increase of 0.00165$\pm0.000006\, {\rm s\, d}^{-1}$, one of the fastest spin-down rates ever recorded. The rapid spin-down coupled with an X-ray luminosity several orders of magnitude lower than the available spin-down power, strongly indicates that the system is in a magnetic ``propeller'' state, namely the rotational energy powers the system's X-ray luminosity. We measured a previously unknown orbital period of 1.357$\pm0.005\,{\rm days}$ for V1716 Sco. If the X-ray flux modulation with a period of 77.9 s detected in outburst for this nova is due to the rotation of an strongly magnetized white dwarf as in other novae with similar modulations of the supersoft X-ray source in outburst, the system is in a parameter space that challenges standard models of cataclysmic variable evolution. For V1674 Her, which has already been classified as an intermediate polar (IP), we confirm the known spin period of 501.328$\pm0.024\,{\rm s}$ and the orbital period of $0.15293 \pm 0.00004$ days, suggesting that the spin modulation was also the root cause of the periodicity in X-rays in outburst, and that the WD atmosphere in the supersoft X-ray phase was not thermally homogeneous. Our results highlight the power of high-cadence, continuous observations in revealing extreme and unexpected characteristics of accreting white dwarfs.

We use martingale embeddings to prove a central limit theorem (CLT) for one-dimensional projections of high-dimensional random vectors in $\{-1,1\}^n$ satisfying a Poincaré inequality. We obtain a non-asymptotic error bound involving two-point and three-point functions for the CLT in 2-Wasserstein distance. We present three illustrative applications: Ising model with finite-range interactions, ferromagnetic Ising model under the Dobrushin condition, and the Sherrington-Kirkpatrick spin glass model at sufficiently high temperature. In all the examples, we allow heterogeneous external fields.

Global digital platforms are software systems designed to serve entire populations, with some already serving billions of people. We propose atomic transactions-based multiagent transition systems and protocols as a formal framework to study them; introduce essential agents -- minimal sets of agents the removal of which makes communication impossible; and show that the cardinality of essential agents partitions all global platforms into four classes: 1. Centralised -- one (the server) 2. Decentralised -- finite $>1$ (bootstrap nodes) 3. Federated -- infinite but not universal (all servers) 4. Grassroots -- universal (all agents but one) Our illustrative formal example is a global social network, for which we provide centralised, decentralised, federated, and grassroots specifications via multiagent atomic transactions, and prove they all satisfy the same basic correctness properties, yet have different sets of essential agents as expected. We discuss informally additional global platforms -- currencies, ``sharing economy'' apps, AI, and more. While this may be the first formal characterisation of centralised, decentralised, and federated global platforms, grassroots platforms have been defined previously, using two incomparable notions. Here, we prove that both definitions imply that all agents are essential, placing grassroots platforms within the broader formal context of all global platforms. This work provides the first mathematical framework for classifying any global platform -- existing or imagined -- by providing a multiagent atomic-transactions specification of it and determining the cardinality of the minimal set of essential agents in the ensuing multiagent protocol. It thus provides a unifying mathematical approach for the study of global digital platforms, perhaps the most important class of computer systems today.

We investigate the stabilisation of nominally linear-affine switched systems with uncertain Lipschitz nonlinearities under dwell-time constraints, using a sampled-data switching law based on a state observer. We design the switching law based on Lyapunov-Metzler inequalities, accounting for the sampled-data output measurements, and we derive time-dependent LMI conditions for global asymptotic stability (or, in the presence of switching affine terms, ultimate boundedness) of the resulting closed-loop system. We obtain an estimate of the average quadratic cost and a bound on its maximum deviation from the actual cost. Moreover, we discuss the feasibility of the derived LMIs. Specifically, we show how the observer gains can be incorporated into the matrix inequalities, provide equivalent reduced-order LMI conditions, and prove that the time dependence of the LMIs can be removed by discretising on a finite grid. Numerical examples, including practical applications to real-world engineering scenarios in power systems, illustrate our theoretical results and compare them with an existing approach for output-feedback stabilisation of switched systems, subject to sampled-data measurements

We study the formation and evolution of stellar halos (SHs) around bright central galaxies (BCGs), focusing on their scaling relations, colors, and metallicities across cosmic time, and compare model predictions with ultra--deep imaging data. We use the semianalytic model \textsc{FEGA25}, applied to merger trees from high--resolution dark matter simulations, including an updated treatment of intracluster light (ICL) formation. SHs are defined as the stellar component within the transition radius, linked to halo concentration. Predictions are compared with observations from the VST Early-type GAlaxy Survey (VEGAS) and Fornax Deep Survey (FDS). The SH mass correlates strongly with both BCG and ICL masses, with tighter scatter in the SH--ICL relation. The transition radius peaks at 30--40 kpc nearly independent of redshift, but can reach $\sim400$ kpc in the most massive halos, after z=0.5. SHs and ICL show nearly identical color distributions at all epochs, both reddening toward $z=0$. At $z=2$, SHs and the ICL are $\sim0.4$ dex more metal--poor than BCGs, but the gap shrinks to $\sim0.1$ dex by the present time. Observed colors are consistent with model predictions, while observed metallicities are lower, suggesting a larger contribution from disrupted dwarfs. SHs emerge as transition regions between BCGs and the ICL, dynamically and chemically coupled to both. Their properties depend on halo concentration, ICL formation efficiency, and the progenitor mass spectrum. Upcoming wide--field photometric and spectroscopic surveys (e.g. LSST, WEAVE, 4MOST) will provide crucial tests by mapping structure, metallicity, and kinematics in large galaxy samples.

This work presents a predictive two-point statistical closure framework for turbulence formulated in physical space. A closure model for ensemble-averaged, incompressible homogeneous isotropic turbulence (HIT) is developed as a starting point to demonstrate the viability of the approach in more general flows. The evolution equation for the longitudinal correlation function is derived in a discrete form, circumventing the need for a Fourier transformation. The formulation preserves the near-exact representation of the linear terms, a defining feature of rapid distortion theory. The closure of the nonlinear higher-order moments follows the phenomenological principles of the Eddy-Damped Quasi-Normal Markovian (EDQNM) model of Orszag (1970). Several key differences emerge from the physical-space treatment, including the need to evaluate a matrix exponential in the evolution equation and the appearance of triple integrals arising from the non-local nature of the pressure-Poisson equation. This framework naturally incorporates non-local length-scale information into the evolution of turbulence statistics. Verification of the physical-space two-point closure is performed by comparison with direct numerical simulations of statistically stationary forced HIT and with classical EDQNM predictions for decaying HIT. Finally, extensions to inhomogeneous and anisotropic turbulence are discussed, emphasizing advantages in applications where spectral methods are ill-conditioned, such as compressible flows with discontinuities.

We prove Kantorovich duality for a linearized version of a recently proposed non-quadratic quantum optimal transport problem, where quantum channels realize the transport. As an application, we determine optimal solutions of both the primal and the dual problem using this duality in the case of quantum bits and distinguished cost operators, with certain restrictions on the states involved. Finally, keeping the same restrictions regarding the states involved, we use this information on optimal solutions to give an analytical proof of the triangle inequality even for the square of the induced quantum Wasserstein divergences.

An algebraic interpretation of matrix-valued orthogonal polynomials (MVOPs) is provided. The construction is based on representations of a ($q$-deformed) Lie algebra $\mathfrak{g}$ into the algebra $\operatorname{End}_{M_n(\mathbb{C})}(M)$ of $M_n(\mathbb{C})$-linear maps over a $M_n(\mathbb{C})$-module $M$. Cases corresponding to the Lie algebras $\mathfrak{su}(2)$ and $\mathfrak{su}(1, 1)$ as well as to the $q$-deformed algebra $\mathfrak{so}_q(3)$ at $q$ a root of unity are presented; they lead to matrix analogs of the Krawtchouk, Meixner and discrete Chebyshev polynomials.

Magnesium (Mg) isotopic ratios offer valuable insights into stellar nucleosynthesis and Galactic chemical evolution, particularly in distinguishing contributions from supernovae and asymptotic giant branch (AGB) stars. These isotopes are accessible via MgH molecular features in cool stellar atmospheres, though their measurement remains challenging across spectral types. We assess the reliability of MgH spectral regions for extracting magnesium isotopic ratios ($^{24}$Mg, $^{25}$Mg, $^{26}$Mg) in stars from M to G types and evaluate consistency with nucleosynthetic expectations. Using spectrum synthesis, we applied an analysis pipeline, validated by three well-studied reference stars, to a sample of five additional dwarf and giant stars. Individual MgH band regions were analysed for sensitivity to isotopic variation. Europium (Eu) and barium (Ba) abundances were also measured to explore correlations with Mg isotopic ratios as r- and s-process proxies. Of ten previously studied MgH wavelength regions, we identify seven as most reliable for isotopic analysis; others showed limited sensitivity across stellar types. Derived Mg isotope ratios ($^{24}$Mg, $^{25}$Mg, $^{26}$Mg) include: HD 11695-81:7:12; HD 18884-81:7:12; HD 18907-69:9:23; HD 22049-71:16:13; HD 23249-66:13:22; HD 128621-67:17:16; HD 10700-78:10:12; HD 100407-65:10:25. Comparison of Eu abundances with Mg isotopes reveals strong correlations, particularly with ($^{24}$Mg, which is predominantly produced by hydrostatic $α$-capture in massive stars, a process preceding the r-process responsible for Eu. In contrast, Ba shows no significant correlation with $^{25}$Mg or $^{26}$Mg, despite their shared s-process origin. Our results demonstrate that selected MgH regions can reliably measure Mg isotopes in cool stars, providing a reproducible framework for future studies of stellar nucleosynthesis and Galactic chemical evolution.