Scientists agree about the urgency of the problem of climate change. Most citizens, however, pay little attention to gradually increasing temperature levels. Growing numbers of natural disasters in the world might then play a fundamental role as the key signal to alert humanity to the severity of the problem of the changing climate. But is that potential mechanism working? In this empirical examination (N>2 million over three decades in 93 countries), we show for the first time that a typical person's happiness and life satisfaction is barely affected by natural disasters in their region. Yet these are the individuals -- as opposed to the minority literally flooded or literally badly affected by hurricanes -- who effectively shape how governments act. This study's ``psychological near-irrelevance'' result is deeply troubling.

The manifestly gauge-invariant and non-perturbatively complete lattice formulation of the weak interactions and the Brout-Englert-Higgs effect is connected to the usual perturbative description in phenomenology via the Fröhlich-Morchio-Strocchi mechanism. However, slight differences between the two have been observed, which can potentially be accounted for by augmenting perturbation theory. We report on our ongoing lattice investigations of these additional effects using a setup with two generations of leptons coupled vectorially to the gauge-Higgs system. We explore the spectrum, inner structure in terms of weak (quasi-)PDFs, and spectral functions of the system to eventually compare cross sections to experimental results.

We prove that $\frac{\log n}{n}$ is the sharp threshold for universality of the distribution of cokernels of random matrices over $\mathbb{Z}_p$. More precisely, let $α_n = \frac{c\log n}{n}$ for a constant $c>0$ and let $A(n)$ be an $α_n$-balanced random matrix over $\mathbb{Z}_p$. For non-symmetric, symmetric, and alternating matrix models, we prove that if $c>1$, then the limiting distribution of the cokernel of $A(n)$ coincides with the universal distribution of the corresponding symmetry type, whereas universality fails at the critical scale $c=1$. This improves earlier universality results, which required $α_n \gg \frac{\log n}{n}$, to the optimal threshold. As an application, we generalize the universality result for Sylow $p$-subgroups of sandpile groups of Erdős-Rényi random graphs to a broader class of Erdős-Rényi graph sequences. Our approach is based on a unified framework that simultaneously treats all symmetry types of random matrices as well as the random graph model, rather than handling each case separately.

The synchronized activity of neuronal populations can lead to pathological over-synchronization in conditions such as epilepsy and Parkinson disease. Such states can be desynchronized by brief electrical pulses. But when the underlying oscillating system is not known, as in most practical applications, to determine the specific times and intensities of pulses used for desynchronizaton is a difficult inverse problem. Here we propose a desynchronization scheme for neuronal models of bi-variate neural activity, with possible applications in the medical setting. Our main argument is the existence of a peculiar point in the phase space of the system, the centroid, that is both easy to calculate and robust under changes in the coupling constant. This important target point can be used in a control procedure because it lies in the region of minimal return times of the system.

We characterize all pairs $(β,n),(β^\prime,m)$ such that the alternate $(β,n)$ and $(β^\prime,m)$-transformations $K_{(β,n)}$ and $K_{(β^\prime,m)}$ have the same absolutely continuous invariant measure, where $K_{(β,n)}(i,x)=(i+1 \mod 2 ,T_i(x))$ with $i\in\{0,1\}$, $T_0(x)=T_β(x)=βx \mod 1$, $T_1(x)=T_n(x)=nx\mod 1$ with $β>1$ real and $n\geq 2$ an integer.

Filaments are ubiquitous throughout the Galaxy. Massive star formation is often observed in hub-filament systems, where multiple filaments appear to be interconnected and merging. Filament-filament collisions are therefore a likely triggering mechanism for massive star formation. We derive basic physical properties of filament-filament collisions, such as the collision cross section (CCS), the hub mass, and its mass function, based on a simple cylindrical filament model. We assume a cylindrical filament with length $2p$, full width $2q$, and line-mass $λ_0$, and consider the CCS between two identical filaments. The collision is specified by three vectors: the directions of the colliding filaments ($n_1$ and $n_2$) and the direction of the relative velocity between the two filaments ($n_v=v/|v|$). For the thin filament, $p\gg q$, the CCS is expressed as $S=4p^2|n'_1\times n'_2|$, where $n'_1$ and $n'_2$ represent the directional vectors projected onto a plane perpendicular to the relative velocity $n_v$. As the angle between $n'_1$ and $n'_2$ becomes smaller, the cross section proportional to $p\cdot q$ becomes relatively important. We propose a simple model in which the hub mass is estimated by the overlapping portion of the two colliding filaments. The hub mass function is derived using the CCSs and the geometrically estimated overlapping mass. When the directions and relative velocities of the filaments are isotropically distributed, the mass function expected from a single species of filaments fits well to a power law and the power exponent is $γ_M\simeq -2.96$ ~ $-3.78$. The power exponent of the global hub mass function is the same as that of the line-mass distribution function, $γ_λ\simeq -1.5$. This means that a massive hub is formed by the collision of two massive filaments.

Via a fixed point argument, we construct solitary waves for the two-dimensional Zakharov system that travel with any small speed $c \in \mathbb{R}^2$. Moreover, we investigate their asymptotic behavior.

Apps such as Firechat and Bridgefy have been used during recent protests in Hong Kong and Iran, as they allow communication over ad-hoc wireless networks even when internet access is restricted. However, these apps do not provide sufficient protection as they do not achieve forward secrecy in unreliable networks. Without forward secrecy, caught protesters' devices will disclose all previous messages to the authorities, putting them and others at great risk. In this paper, we introduce FoSAM, the first protocol to provide proven anonymous and forward secret messaging in unreliable ad-hoc networks. Communication in FoSAM requires only the receiver's public key, rather than an interactive handshake. We evaluate the performance of FoSAM using a large-scale simulation with different user movement patterns, showing that it achieves between 92% and 99% successful message delivery. We additionally implement a FoSAM prototype for Android.

The widespread adoption of randomized controlled trials (A/B Tests) for decision-making has introduced a pervasive "Winner's Curse": experiments selected for launch often exhibit upwardly biased effect estimates and invalid confidence intervals. This selection bias leads to over-optimistic impact projections and undermines decision-making, particularly in low-power regimes. We propose Bayesian Hybrid Shrinkage (BHS), an empirical Bayes (EB) framework that leverages data-driven priors to mitigate selection bias and provides accurate uncertainty quantification. Unlike traditional EB methods that apply uniform shrinkage, BHS introduces an experiment-specific "local" shrinkage factor that incorporates individual experiment characteristics, improving robustness against prior misspecification. We also derive a closed-form inference strategy designed for high-throughput production environments. Extensive simulations and real-world evaluations at Meta Platforms demonstrate that BHS outperforms existing methods in terms of bias reduction and interval coverage, even under substantial violations of modeling assumptions.

Magnonic excitations are investigated in chromium oxychloride (CrOCl), a van der Waal (vdW) antiferromagnet prone to a multitude of magnetic phase transitions, with absorption experiments in a broad continuous energy range. At low magnetic fields, the magnon spectra show a strong bi-axial anisotropy and inform on the relative weights of the effective exchange coupling and the system anisotropies. As the magnetic field increases, magnons characteristic of a canted phase are first observed, with peculiarities attributed to in-plane anisotropies and magnon-magnon coupling. Subsequently, a hysteretic magnon spectrum appears as the system transitions to a ferrimagnetic state, with two new magnon branches partly coexisting with the lower energy canted phase branch, indicating the formation of spatially separated magnetic phases. Further changes in the magnon spectrum in higher magnetic fields accompany transitions to the different canted magnetic phases previously reported. Our experiments show that competing exchange interactions and ground states broaden the options to generate different kinds of magnonic excitations in the same vdW material upon the variation of external parameters.

A Monte Carlo generator of high energy cosmic ray interactions, relying on a very basic and transparent theoretical formalism, in the framework of the Reggeon Field Theory, is presented. The main motivation for our work is to provide a new cosmic ray interaction model characterized by relatively transparent physics, sufficient parameter freedom, and a high computational efficiency, which can be easily managed by external users, including a re-tuning of the model parameters. Such a model can be used for studying potential modifications of the interaction treatment, necessary for describing particular sets of data on extensive air showers initiated by high energy cosmic rays, at a microscopic level, thereby keeping a consistency with general restrictions, like the unitarity, energy-momentum and charge conservation, Lorentz and isospin invariance. Importantly, this should allow one to study a compatibility of such modifications with relevant accelerator data. The model results for particle production and for basic extensive air shower characteristics are presented and discussed.

We review the construction of locally finite two-loop amplitude integrands for photoproduction via quark annihilation, presented in arXiv:2509.07805. Building on established techniques for off-shell colorless production, we extend this local subtraction framework to handle transient singularities arising from real outgoing photons. These singularities manifest only at the integrand level, yielding finite contributions upon integration. In these proceedings, we provide the explicit construction of ultraviolet counterterms that satisfy necessary Ward identity cancellations, ensuring that the integrand is rendered integrable. This work provides a locally finite amplitude integrand that is ready for numerical integration in momentum space. Furthermore, it establishes the foundation for extending local subtraction frameworks to processes involving final-state jets.

Vertical gas flows, such as winds and meridional circulations, are natural outcomes of protoplanetary disk processes and play a critical role in the earliest stages of planet formation. We analyze vertical gas motions in 14 disks as part of the exoALMA Large Program, focusing on the 12CO J=3-2 and 13CO J=3-2 emission lines. Using discminer to model the Keplerian velocity field, we extract line-of-sight velocity residuals and measure the radial and vertical components of the gas motion. Vertical motions are detected in most disks. Two types of patterns emerge: (1) oscillatory up/down flows, likely linked to instabilities, and (2) transitions from downward to upward motions that we interpret as the base of a disk wind. In most cases, the velocity amplitudes are of a few tens of m/s. Two disks, however, MWC758 and CQ Tau, show two spiral velocity features in their residual maps, red- and blue-shifted, which we interpret as vertical velocities reaching up to 350 m/s (0.7 Cs), consistent with gas motion in eccentric disks. Fast upward motions (up to 500 m/s; 1.8 Cs) is also detected in the outer disk of MWC758. Synthetic observations from (magneto)hydrodynamic simulations validate the reliability of our method. Although strong molecular winds appear to be relatively rare in 12CO and 13CO, our study shows that, when traced by deep high spectral resolution line data, protoplanetary disks exhibit ubiquitous vertical flows. However, their overall velocity structure is highly complex, preventing to identify a coherent, dominant physical mechanism driving the vertical motions across all disks, thus requiring further theoretical investigation.

The solar wind flow creates a wake behind any spacecraft immersed in it. We study the properties of this wake using the spherical electrostatic probes of the Electric Fields and Waves (EFW) instruments on the Cluster satellites. The satellites spin in a plane inclined only a few degrees with respect to the ecliptic plane. The solar wind is often close to this plane, so each probe (44 m away from the spin axis) passes through the wake once every spin period (around 4 s), thereby sampling a cut of the wake electrostatic potential structure. The signature of the wake is clearly seen in the data as a pulse with an amplitude typically of a few tenths of a volt. We present statistics of the wake signatures as well as detailed examples, compare to solar wind parameters, and show a method to remove the wake signature from the electric field measurements.

Auger electron spectroscopy, a way of characterizing electronic structure through core-level decay processes, is widely used in materials characterization; however direct calculation from molecular geometry requires accurate treatment of many excited states, posing a challenge for classical methods. We present a hybrid quantum-classical workflow for calculating Auger spectra that combines the generative quantum eigensolver (GQE) for ground-state preparation, the quantum self-consistent equation-of-motion method for excited-state calculations, and the one-centre approximation for Auger transition rates. GQE uses a GPT-2 model to generate quantum circuits for ground-state optimization, allowing our workflow to benefit from HPC parallelization and GPU-acceleration for favourable scaling with system size. We demonstrate the validity of our workflow by calculating the Auger spectrum of water with the STO-3G basis set and demonstrating qualitative and quantitative agreement with spectra obtained using completely classical full configuration interaction calculations, from the computational literature, and from the experimental literature. We also find that for water, substituting the variational quantum eigensolver (VQE) for GQE results in near-identical spectra, but that the ground state estimator generated by GQE contains about half the total gate count as that generated by VQE.

A purely transmissive Huygens' metasurface model under plane-wave illumination is used to derive circuit parameters describing a constituent unit cell, such that diverse refraction angles are attained at two distinct frequency bands. Various levels of accuracy of the circuit description approaching the analytical are possible by constraining certain numbers of parameters. This theoretical study is then tested by calculating the exact formulas of the two representations for the various strategies proposed. By using simulations of a candidate unit-cell, we then examine whether such circuit parameters correspond to rudimentary versions of the geometry of a so-called parallel 'dogbone' structure. A device of this type is intended as dual-band (dichroic), dual-angle beam refractor diverting an incoming beam at different directions in two different bands without reflections.

Perovskite solar cells (PSCs) have experienced a remarkable rise in power conversion efficiency (PCE) over the past 15 years, positioning them as a promising alternative or complement to silicon for large-scale photovoltaic deployment. However, beyond scalable fabrication, operational stability remains a major bottleneck for commercialization. Reliable and rapid methods to assess device health and degradation mechanisms - ideally compatible with field applications - are therefore essential. We present a deep-learning framework to estimate efficiency retention, $R_\mathrm{PCE}=\mathrm{PCE}_t/\mathrm{PCE}_0$, directly from multimodal luminescence imaging acquired during device aging. Each training sample includes electroluminescence (EL), open-circuit photoluminescence (PLoc), and short-circuit photoluminescence (PLsc) images at an aged state, together with device-specific reference images at $t=0$. This design enables the model to learn spatially resolved degradation patterns relative to the pristine condition. The dataset was collected over 5-70 hours using an automated, in-house measurement platform. We introduce LumPerNet, a compact convolutional neural network that regresses $R_\mathrm{PCE}$ from stacked multimodal image tensors, and benchmark it against an intensity-only multilayer perceptron baseline. Using a leakage-aware protocol with device-level hold-out testing and four-fold cross-validation, restricted to $R_\mathrm{PCE}\in[0.8,1.2]$, LumPerNet achieves substantially improved and more robust performance (MAE -23.4%, RMSE -25.6%, $R^2$ +0.417). Ablation studies highlight the importance of complementary physical contrast across modalities for generalization. Overall, this work establishes a reproducible pipeline linking automated luminescence imaging to electrical labels, enabling accelerated stability testing and non-invasive degradation monitoring in PSCs.

An axion-like inflaton coupled to non-Abelian gauge bosons provides a compelling microphysical framework for warm inflation. Starting even from cold initial conditions, in these systems, sphaleron heating may generate thermal friction sufficient to sustain finite temperatures throughout the inflationary epoch. Insisting on shift-symmetric potentials, in this work we revisit the viability of these scenarios under the designation of Minimal Warm Inflation. We examine both observational constraints and model-building limitations on models with a hierarchy between the decay constants appearing in the friction rate and in the inflaton potential. We conclude that the popular clockwork mechanism cannot generate the required hierarchy; however, partial-wave unitarity bounds admit effective descriptions that remain consistent with observations.

The blazar PKS 0903-57 has recently been classified as a flat spectrum radio quasar at a redshift of $z=0.2621$. In March and April 2020, Fermi-LAT and AGILE reported tremendous activity in high-energy $γ$ rays with the flux increasing by $\sim$2 orders of magnitude compared to quiescence. The flare was observed with H.E.S.S. in very-high-energy $γ$ rays for six nights with a total observation time of 13.1 h, resulting in the discovery of PKS 0903-57 in this energy band with an average flux of $1.5\times 10^{-10}$ ph cm$^{-2}$s$^{-1}$ above an energy threshold of $\sim 180$ GeV corresponding to $60\%$ of the Crab Nebula flux above the same threshold. The very-high-energy $γ$-ray flux was strongly variable. X-ray and optical data were collected with Swift and ATOM, and also indicate significant variability. The observed multiwavelength flux and spectral variability during the H.E.S.S. observation window suggest variability time scales on the order of a few hours and reveal complex correlation patterns. The lack of absorption beyond that of the extragalactic background light in the $γ$-ray domain suggests that the emission region was located outside of the broad-line region. A leptonic one-zone modeling of the six H.E.S.S. observation nights using the dusty torus as seed photons for the inverse-Compton scattering, results in a low magnetization of the emission region. This implies that shock acceleration is likely the main driver during the event.

We investigate optimal stopping problems for systems driven by the Brownian sheet. Our analysis is divided into two parts. In the first part we derive explicit solutions to two optimal stopping problems for the exponentially discounted Brownian sheet. The first problem consists in determining the optimal two-parameter first hitting point tau = (tau1,tau2) maximizing E[exp(-rho tau1 tau2) h(B(tau1,tau2))], where rho > 0 is a discount factor and h is a reward function. Restricting attention to first hitting points of levels, we obtain a closed-form characterization of the optimal stopping threshold. In particular, for linear rewards h(y)=y the optimal level is y_hat = (2 rho)^(-1/2). The second problem concerns optimal stopping of the integrated discounted Brownian sheet with payoff E[int_0^{tau1} int_0^{tau2} exp(-rho t x) B(t,x) dt dx]. We show that the optimal first hitting level is strictly positive and give an explicit representation of the value function in terms of the exponential integral function. The optimal threshold is characterized as the unique solution of a nonlinear equation derived from a Laplace transform identity for the product tau1 tau2. In the second and main part of the paper we develop a potential theoretic framework for two-parameter optimal stopping problems associated with stochastic partial differential equations driven by the Brownian sheet, proving that the value function is the least superharmonic majorant of the reward and establishing existence of optimal stopping points in the plane.

We develop variational continuous unitary transformations (VCUTs), which integrate Wegner-Wilson flow equations with tensor network techniques to approximately diagonalize many-body localized (MBL) Hamiltonians. The diagonalizing unitary is represented as a matrix product operator whose bond dimension controls the accuracy. For the disordered Heisenberg chain, VCUTs accurately reproduces the full spectrum across the ergodic-to-MBL crossover at small system sizes and scales to $L = 48$ sites. Beyond eigenenergies, the method can track the spatial entanglement structure of the diagonalizing unitary $U(l)$ at each flow step, enabling identification of local integrals of motion deep in the MBL phase.

The realization of a scalable quantum internet requires efficient light-matter interfaces that map stationary qubits onto photonic carriers for long-distance transmission. A central challenge is the generation of entangled photons simultaneously compatible with single-emitter transitions and low-loss telecom fiber infrastructure. Spontaneous parametric down-conversion in integrated photonic platforms offers a promising route toward this goal. Among available material systems, AlGaAs is particularly attractive due to its large second-order nonlinearity and strong potential for monolithic integration. However, engineering the spectral and spatial properties of the generated quantum states requires the simultaneous optimization of numerous geometric and material parameters, a task remaining computationally demanding for conventional numerical approaches. To address this challenge and enable rapid and high-fidelity modeling of complex nonlinear photonic devices, we develop an inverse-design framework based on neural network surrogate models. Using this readily extendable method, we design a transversely pumped AlGaAs waveguide microcavity that produces polarization-entangled photon pairs in distinct spatial modes and frequency channels, one at 1092 nm, resonant with a $^{88}\text{Sr}^{+}$ transition, and the other at 1550 nm in the telecom C-band. This device establishes a direct photonic interface between trapped-ion qubits and long-haul fiber networks, providing a scalable pathway toward hybrid quantum network architectures.

We introduce a functional renormalization group framework formulated directly in the Floquet steady-state that systematically incorporates frequency-dependent interaction effects. By retaining the frequency structure of the two-particle vertex up to second order in interaction strength, our approach provides controlled access to dynamical response functions and nonequilibrium transport in driven, interacting systems. Using the periodically driven single-impurity Anderson model as a paradigmatic example, we benchmark our results against state-of-the-art Floquet Green's function methods and find quantitative agreement for finite-frequency observables up to intermediate interaction strengths. Remarkably, we also show that static properties are often captured reliably by much simpler approximations, suggesting practical pathways for modeling driven quantum materials. Finally, we demonstrate that although periodic driving of the dot strongly broadens the Kondo resonance through inelastic scattering, it leaves the many-body Kondo cloud largely intact. This robustness suppresses Floquet replicas of the Kondo peak and leads to a partial persistence of Kondo pinning, highlighting the resilience of emergent many-body correlations under local periodic driving.

In this paper, we elucidate the geometry of Stein's method of moments (SMoM). SMoM is a parameter estimation method based on the Stein operator, and yields a wide class of estimators that do not depend on the normalizing constant. We present a canonical decomposition of an SMoM estimator after centering the score matching estimator, which sheds light on the central role of the score matching within the SMoM framework. Using this decomposition, we construct an SMoM estimator that improves upon the score matching estimator in the asymptotic variance. We also discuss the connection between SMoM and the Wasserstein geometry. Specifically, using the Wasserstein score function, we provide a geometrical interpretation of the gap in the asymptotic variance between the score matching estimator and the maximum likelihood estimator. Furthermore, it is shown that the score matching estimator is asymptotically efficient if and only if the Fisher score functions span the same space as the Wasserstein score functions.

This work introduces a port-Hamiltonian (PH) model for constrained mechanical systems, which is directly derived from the Lagrangian equations of motion. The present PH framework incorporates a singularity-free director representation of rigid body rotations, resulting in constant mass matrices. It is shown that the power-preserving interconnection of PH rigid-body subsystems is mathematically equivalent to the classical description of ideal joints using kinematic pairs. This establishes a PH multibody dynamics framework that is consistent with traditional modeling paradigms. Notably, the PH structure of the governing index-2 differential-algebraic equations enables the application of an implicit, structure preserving midpoint time integration. The proposed scheme is able to satisfy both the balance laws for total energy and angular momentum as well as the position-level constraints. These properties make the proposed method remarkably robust and enable stable long-term simulations. Furthermore, a variationally derived index-reduction strategy is incorporated that enforces velocity-level constraints in addition to position-level constraints while preserving the port-Hamiltonian structure. Numerical examples illustrate the favorable properties of the proposed formulation, which is well-suited for energy-based control design.

Cells exert traction forces on compliant substrates and can induce surface instabilities that appear as characteristic wrinkling patterns. Here, we develop a mechanical description of cell-induced wrinkling on soft substrates using a thin film elastic framework based on the Föppl-von Kármán equations coupled to a phase-field model of a single cell. We model in-plane contractile stresses driven by cellular activity and study how their magnitude, spatial distribution, and symmetry determine the onset of wrinkling and the resulting pattern selection. The theory predicts transitions between distinct morphologies, such as radial, circumferential, and anisotropic wrinkle arrangements, and provides scaling relations for wrinkle wavelength and amplitude as functions of elastic parameters and imposed cellular forcing. We compare these predictions with available experimental observations of cell-driven wrinkling on compliant gels and find good agreement for both qualitative pattern classes and quantitative wavelength trends. Our results offer a minimal modelling framework to interpret wrinkling assays and connect observed surface patterns to underlying cellular forces.

Existing convergence of distributed optimization methods in non-Euclidean geometries typically rely on kernel assumptions: (i) global Lipschitz smoothness and (ii) bi-convexity of the associated Bregman divergence function. Unfortunately, these conditions are violated by nearly all kernels used in practice, leaving a huge theory-practice gap. This work closes this gap by developing a unified analytical tool that guarantees convergence under mild conditions. Specifically, we introduce Hessian relative uniform continuity (HRUC), a regularity satisfied by nearly all standard kernels. Importantly, HRUC is closed under concatenation, positive scaling, composition, and various kernel combinations. Leveraging the geometric structure induced by HRUC, we derive convergence guarantees for mirror descent-based gradient tracking without imposing any restrictive assumptions. More broadly, our analysis techniques extend seamlessly to other decentralized optimization methods in genuinely non-Euclidean and non-Lipschitz settings.

This paper presents the first bit error rate (BER) analysis of a pinching-antenna (PA)-based non-orthogonal multiple access (NOMA) communication system. The PA is assumed to be able to be placed anywhere along the waveguide and serves two NOMA user equipment (UEs) in both uplink (UL) and downlink (DL) scenarios. Exact closed-form expressions for the average BER of each user are derived under practical imperfect successive interference cancellation (SIC). These expressions are then used to optimize the PA location for minimizing the overall average BER of both UEs. In the UL case, the interference between the users' channels introduces phase-dependent fluctuations in the BER cost function, making it highly non-convex with many local extrema. To address this challenge, a smoothing technique is applied to extract the lower envelope of the BER function, effectively suppressing ripples and enabling a reliable identification of the global minimum. In the DL case, a joint optimization of the PA location and NOMA power allocation coefficients is proposed to minimize the average BER. Simulation results verify the accuracy of the analytical derivations and the effectiveness of the proposed optimization methods. Notably, the UL results demonstrate that an optimally positioned PA can create the required received power difference between two equally powered UEs for reliable power-domain NOMA decoding under imperfect SIC.

The focus of this study is on exploring some qualitative properties of solutions to a class of semilinear elliptic problems in bounded domains, where the boundary conditions depend non-locally on the unknown solution at specified interior points and its integral. The primary approach integrates a fixed-point argument with refined asymptotic estimates to establish the existence and structure of solutions. Furthermore, the maximum principles are established under practical nonlocal-type boundary conditions.

Synthesizing fully developed three-dimensional turbulent velocity fields remains a long-standing problem in fluid mechanics and an open challenge for generative modeling. The difficulty arises from the coexistence of extreme dimensionality, multiscale rough fluctuations and strong intermittency, together with exact physical constraints such as incompressibility and zero-mean momentum. We propose a physics-constrained diffusion model (PCDM) in which these \emph{a priori} constraints are incorporated directly into the generative dynamics. Using rotating turbulence as a stringent benchmark, we show that the proposed framework enables stable and statistically faithful synthesis of inertial-range three-dimensional turbulent velocity fields at medium resolution, accurately reproducing anisotropic energy spectra, intermittency statistics, and physical constraints. By contrast, standard denoising diffusion probabilistic models without such constraints exhibit multiscale statistical deviations, violations of physical consistency, and substantially slower training convergence. These findings point to broader implications for generative modeling of high-dimensional complex systems under physical constraints.