We present new convergence estimates for the iterated penalty method applied to structure-preserving discretizations of linear generalized saddle point systems. The method may be viewed as an Uzawa iteration on an augmented Lagrangian formulation of the system. As a by-product, we obtain sharper stability estimates for penalized/perturbed saddle point problems. Three model finite element applications show agreement with the theory.

We use the high-low method and wavepacket pruning to prove new small-cap decoupling estimates for the moment curve in $\mathbb{R}^4$. As an application, we verify a conjecture of Demeter regarding the $L^{12}$ square-root cancellation of exponential sums associated with the moment curve in $\mathbb{R}^4$. This provides a continuum of square-root cancellation estimates that connects the Vinogradov MVT in $\mathbb{R}^3$ with a result of Bourgain, related to improving the best-known estimate for the Lindelöf hypothesis.

We propose a formally valid machine-learning-assisted global proposal mechanism for Monte Carlo sampling in lattice gauge theory. The construction is based on a coupling-flow update on the SU(2) lattice-link manifold, in which active links are transformed conditionally on a frozen-link background. For fixed frozen links, the proposal is explicitly invertible and preserves the product Haar measure, so it can be embedded into a Metropolis-Hastings correction without requiring an explicit model of the full proposal density. We implement the method in two-dimensional pure SU(2) lattice gauge theory and benchmark it against a baseline local Metropolis algorithm used as a controlled reference kernel. In the present testbed, the learned proposal reproduces the target ensemble within statistical resolution across the tested configurations. In matched local-step comparisons, the learned proposal reproduces the target ensemble at a quality comparable to the baseline, but does not outperform the pure local baseline in the conservative matched-step case examined with seed-level statistics within this proof-of-principle setup. At the same time, a favorable mixed-step hybrid configuration yields a modest improvement in effective sample size per unit runtime. Because the learned transformation remains in a near-identity regime, the present results should be interpreted as a proof-of-principle demonstration of formal correctness and limited, configuration-dependent efficiency gain within a controlled comparison, rather than as evidence of superiority over optimized conventional update schemes. This work provides a concrete foundation for extending machine-learned nonlocal updates to larger lattices and non-Abelian gauge theories relevant to lattice QCD.

We study a family of above-below Ramsey functions $\operatorname{AB}^{(d)}(k)$ defined for sequences of points in $\mathbb R^d$ whose projections to $\mathbb R^{d-1}$ have cyclic order type. The case $d=3$ is the above-below function $\operatorname{AB}(k)$ that was first introduced by Pohoata and Zakharov in their work on the Erdős-Szekeres problem in $\mathbb{R}^{3}$. We prove the sharp estimate \[ \operatorname{AB}(k)=2^{2^{Θ(k)}}, \] and, more generally, show that $\operatorname{AB}^{(d)}(k)$ is closely related to the higher-order cup-cap function of Eliáš and Matoušek and the monotone Ramsey numbers of Balko.

We show that Hall viscosity in lattice bands is governed by a band-projected electric quadrupole encoded within the quantum geometry: Berry curvature sets the projected-coordinate algebra, while the quantum metric determines the quadrupolar spread of a wave packet. The same structure enters the quadratic wave-vector coefficient of the nonlocal Hall conductivity, yielding a lattice viscosity-conductivity relation. In ideal bands, the deviation from the Landau-level form is quantified by Berry curvature fluctuations. Our results establish the nonlocal Hall response as an electrical signature of the quantum geometry underlying Hall viscosity and as a transport diagnostic of geometric idealness.

Let $f,g\in\mathbb{C}[z]\setminus\mathbb{C}$ and $c\in\mathbb{C}[z]$. Suppose that $\mathrm{deg}(c)=1$ if $\mathrm{deg}(f)=\mathrm{deg}(g)=1$. Using the theory of Presburger arithmetic, we prove that the rank-two recurrence set \[S_{f,g,c}^2:=\left\lbrace(m,n)\in\mathbb{Z}_{\geq0}^2\colon \existsλ\in\mathbb{C}, f^{\circ m}(λ)=g^{\circ n}(λ)=c(λ)\right\rbrace\] is semi-linear. This is a generalization of a theorem of Yang and Zhong for the case $m=n$. We also obtain partial results on recurrence sets for rational maps in the case $m=n$. These results are related to higher-dimensional questions of dynamical Mordell--Lang type of rank $\leq2$.

We present a hybrid forecasting strategy that combines numerical modeling, statistical forecasting, and machine learning methods to predict enhanced bursts of solar activity. These bursts, referred to here as space weather seasons, occur on intermediate timescales of approximately 6 to 18 months. We analyze monthly smoothed sunspot number data from 1878 to 2025 and use Gaussian fitting to identify burst events and characterize their amplitudes and durations. The sunspot number data are divided into training, testing, and forecasting intervals to evaluate both hindcast performance and future predictions. Each solar hemisphere is modeled separately using a seasonal autoregressive integrated moving average approach. This baseline forecast is refined using an asymmetric Gaussian override to represent the rapid rise and gradual decay of burst activity. Burst amplitudes and durations are then estimated using a random forest regression model. The hybrid method successfully reproduces burst timing between November 2024 and May 2025, with the Northern Hemisphere reaching a peak sunspot number of about 70 around March 2025. The next Northern Hemisphere burst is forecast to occur around December 2025, with a slightly lower peak sunspot number of about 60. In contrast, the Southern Hemisphere shows more complex behavior, with multiple burst amplitudes beginning around October 2024 and continuing until October 2025. The dominant Southern Hemisphere burst reaches an amplitude of about 130 in sunspot number, and the next burst is forecast to occur around December 2025. When the hemispheric forecasts are combined, the total sunspot number is found to be mainly controlled by the stronger activity cycle in the Southern Hemisphere.

We present the on-site rotation performance of the first three cryogenic continuously rotating half-wave plate (HWP) polarization modulators for the Simons Observatory small aperture telescopes (SATs). The SATs operate at an altitude of 5200 m in the Atacama Desert in northern Chile, and measure the degree-scale cosmic microwave background polarization to search for primordial $B$-mode polarization. To this end, the SATs employ a 505 mm diameter 50 K cryogenic HWP polarization modulator to suppress atmospheric $1/f$ noise and to mitigate systematic uncertainties. We present methods for reconstructing the rotation of our polarization modulators, enabling detailed evaluations of their rotation angle accuracy, stability, displacements, and vibrations. We achieve the required rotation angle accuracy in more than 99.9% of observations, with a median noise level of 0.16 $μ$rad$\sqrt{\text{s}}$. We also achieve one-dimensional measurement of the rotor displacement with an accuracy of 0.06 $μ\text{m}\sqrt{\text{s}}$. Our results demonstrate the on-site rotation performance, rotation angle reconstruction method, and robustness of the polarization angle modulators. This is a crucial step towards achieving the SAT science goals.

Generalizing motion representation across diverse characters remains challenging due to significant topological variations in skeletal structures across datasets and species, which hinder the development of scalable generative models. To bridge this gap, we propose a Semantic-Aware Topology-Agnostic framework that learns a unified latent manifold shared by disparate species. Unlike methods relying on fixed hierarchies or rigid padding strategies, our approach leverages a semantic modulation mechanism to align functional joint correspondences, thereby decoupling motion from topology. This design enables the construction of a continuous, generative-friendly motion space from large-scale, unaligned raw BVH data. Experiments on human and animal datasets demonstrate that our framework achieves high-fidelity reconstruction and supports downstream text-to-motion tasks. Notably, the model enables zero-shot cross-species retargeting without paired data. Code and demos are available at: https://github.com/zzysteve/SATA

We prove that the Cauchy problem for the model hyperbolic operator in $ \R^{4} $ \[ Q=-D_t^2+2xD_tD_y+D_x^2+x^3D_y^2+D_z^2+z^2D_y^2 \] is not locally solvable at the origin, in the Gevrey $s$ class if $s>6$.

We revisit the three-body problem in quantum mechanics in two and three dimensions, generating both exact eigenvalues and eigenvectors of the Hamiltonian and a series of approximate solutions as calculated with a variety of different schemes to separate heavy ("nuclear") and light ("electronic") particles. We show that, with minimal extra cost, one can go beyond the Born-Oppenheimer approximation by performing electronic structure calculations parameterized by both the nuclear position (${\mathbf X})$ and the nuclear momentum ($\mathbf{P}$), a so-called phase space theory of electronic structure. In particular, we demonstrate that such phase space electronic structure calculations correctly incorporate the non-inertial Coriolis and centrifugal forces felt by electrons in a moving nuclear frame, thus leading to far more accurate eigenenergies and electronic angular momenta than has been possible before. We also demonstrate that our approach naturally incorporates and generalizes the Moody-Shapere-Wilczek magnetic monopole for the non-abelian Berry curvature (now allowing for vibrational motion rather than a diatomic of fixed length). We argue that the resulting approach should be extremely useful for propagating dynamics where angular momentum flows between nuclei and electrons; in particular, if extended to include spin degrees of freedom, the present approach will offer a practical means to study chiral induced spin selectivity through the lens of chiral phonons and coupled nuclear-electronic motion.

The fundamental correspondence between quantum chaotic single-particle systems and random matrix theory is well-understood via periodic orbit theory. In contrast, we show that many-body systems with explicit subsystem structure possess characteristics different from the single-particle theory. We present a periodic orbit theory for many-body systems with well defined semiclassical limit. For this we identify periodic orbit families arising exclusively in the many-body setting and implement a central limit theorem characterizing their correlations. Based on this we demonstrate that spectral correlations in chaotic quantum circuits are characterized by the breaking of individual time translation invariance of periodic orbits in the subsystems into residual synchronous time translations only. This provides a systematic approach to confirming random matrix universality in deterministic many-body systems.

Estimating surface properties such as porosity and grain sizes is key for planning lander missions and landing site selection on icy moons. However, spaceborne instruments do not measure the regolith properties directly: instead, they record proxy measurements such as thermal flux, which are then interpreted through modeling to estimate thermal inertia, porosity, grain size, etc. A striking conclusion from all thermal measurements that probed the uppermost surface (first millimeters) of icy moons is they all show an exceptionally low thermal inertia, ranging from 9 to 20 J.m-2.K-1.s-0.5. This value is orders of magnitude lower than that of bulk hexagonal water ice (2000 J.m-2.K-1.s-0.5) at these temperatures. We demonstrate that a regolith thermally dominated by hexagonal water ice may only achieve such thermal inertia through a combination of extremely high porosity (>80%), small grain radii (<1 mm), and an unconsolidated regolith (minimal contact area between grains), consistent with previous photometry and spectroscopy studies. For the Galilean moons, deeper thermal observations (>1 cm) have revealed higher thermal inertia (>~50 J.m-2.K-1.s-0.5), indicating that the regolith compacts over centimeter scales. Since gravity has no effect on compaction on such scale, we propose three formation scenarios to account for vertical layering: deposition cover, degradation by impactors, and temperature gradient metamorphism. We discuss how monodisperse grains can reach such extreme porosities and provide examples of experimental analogs that could best represent the regolith. We propose that high porosity regolith are favored on icy moons due to the adhesive nature of water ice and their low-gravity environment.

Fast-responding trap states introduced by NO-annealing are suspected to limit the channel mobility of 4H-SiC MOSFETs, yet their microscopic characterization remains challenging because conventional electrical methods are spatially averaged and do not readily isolate such fast processes. Here, we visualize and analyze individual fast-responding trap states at the NO-annealed SiO$_2$/4H-SiC interface using the energy dissipation signal in frequency-modulation atomic force microscopy (FM-AFM), which selectively probes charge-exchange dynamics on sub-$μ$s time scales. Ring-shaped dissipation patterns were observed in the NO-annealed sample but not in the control sample without NO-annealing, indicating that the detected states are associated with nitridation. Spectroscopic measurements were also performed to determine the dependence of energy dissipation on the tip bias and the tip-sample distance. Combined with finite-element electrostatic calculations, this analysis allowed us to determine trap energies relative to the Fermi level, $E_t - E_f$, and revealed that the trap-energy distribution extends toward the interfacial conduction-band edge. These results provide microscopic evidence that NO-annealing generates fast-responding trap states near the SiO$_2$/4H-SiC interface.

Selective targeting of membranes with a specific receptor profile is an ongoing challenge in targeted drug delivery. We investigate the adsorption of copolymers on a multicomponent receptor-covered surface using grand-canonical Monte Carlo simulations and demonstrate that polymers can be designed to target a particular receptor density profile. To achieve this, the ligand profile on the polymers should match the targeted receptor profile, and the ligand--receptor affinity should be inversely proportional to the ligand profile. While the same can be obtained using multivalent nanoparticles, the entropic effects due to polymer conformations significantly enhance the binding selectivity of multivalent polymers compared to nanoparticles. Surprisingly, the ligand distribution on the polymer plays a crucial role, whereas the persistence length does not. The optimal selectivity to the overall receptor concentration is obtained by the Poisson distribution of ligands (random copolymer), whereas the maximal selectivity to a specific receptor profile is obtained by a defined sequence of grouped alternating ligands (regular copolymer). Interestingly, the regular copolymer can become anti-selective when ligands of the same type are in homogenous blocks, showing that specific ligand distribution qualitatively affects the targeting ability. These findings suggest that sequence control is necessary to selectively target a specific density profile of membrane receptors using linear copolymers.

A nonlocal Busenberg-Travis cross-diffusion system for segregating populations is analyzed in a bounded domain with no-flux boundary conditions. The velocities of the species solve a regularized Darcy law, which can be interpreted as a Brinkman equation. Compared to results in the literature, the density-pressure relation is assumed to be nonlinear. The global existence of weak solutions to this system is shown for a broad range of the exponents of the power-law nonlinearity, and the localization limit is proved. The proofs are based on uniform estimates coming from the Tsallis entropy inequality. Due to regularity issues, the original problem is approximated by various schemes, and the de-regularization limits are obtained through compactness arguments.

The effect of oxygen partial pressure during the pre-calcination step (high vacuum, air, nitrogen, and oxygen) on the crystal structure, microstructure, and electrical properties of Na0.5Bi0.465Sr0.02Eu0.005TiO3 oxide-ion-conducting ceramics was systematically investigated. Dense ceramic samples were prepared by a conventional solid-state reaction route under different atmospheres. The results show that the oxygen partial pressure strongly affects Bi volatilization, grain growth, and oxygen-vacancy concentration. The largest average grain size was obtained for the nitrogentreated sample, whereas the oxygen-treated sample exhibited the finest grains but the highest grain-boundary conductivity. X-ray diffraction and Raman spectroscopy indicate that low oxygen partial pressure enhances structural disorder, while high oxygen partial pressure stabilizes the lattice and promotes charge-transfer transitions. Eu3+ photoluminescence further reveals the correlation between local structural evolution and oxygen-vacancy concentration. These findings clarify how oxygen partial pressure regulates grain-boundary behavior and ion-transport mechanisms in NBT-based oxide-ion conductors, providing guidance for optimizing the total conductivity of polycrystalline electrolytes.

Let p<q be coprime integers. Kunz-Waldi semigroups are numerical semigroups containing p and q and contained in <p,q,r>, where 2r=p,q,p+q whichever is even. In this paper, we prove a conjecture on the Betti numbers of the semigroup rings of these semigroups, showing that they coincide with those of the ideal of 2x2 minors of a 2xn generic matrix, where n is the embedding dimension. Moreover, we characterize the Apéry posets of Kunz-Waldi semigroups and determine when their tangent cones are Cohen-Macaulay.

In this article, we investigate the range characterization for the spherical mean transform (SMT) of functions supported in the unit ball. In earlier works, in the case of odd dimensions, a set of differential conditions was obtained, whereas in the case of even dimensions, integral conditions were obtained. We prove that these conditions that are different based on the parity of dimension are, in fact, equivalent in odd dimensions. This equivalence shows that the integral conditions yield a unified simple range characterization for the SMT that is valid in both even and odd dimensions.

Collective modes provide direct fingerprints of quantum matter. We predict a previously unidentified hybrid bulk electromagnetic mode in Weyl semimetals arising from the interplay between the chiral anomaly and the orientation of its associated chiral magnetic response relative to the direction of the wave-vector. When the anomaly-induced chiral magnetic current has a component along the propagation direction, oscillations of valley imbalance hybridize with plasmonic charge oscillations, producing a linearly dispersing mode that undergoes avoided crossing with the bulk plasmon, producing a hybrid bulk excitation absent in ordinary metals. The hybrid mode provides a direct signature of Weyl semimetals and a probe of the chiral anomaly and its associated chiral magnetic effect, with observable features in electron energy-loss spectroscopy. Studying this interplay can uncover various optical and electronic properties of Weyl semimetals.

We extend Newton's problem of minimal resistance to Riemannian surfaces endowed with a geodesic coordinate system, which includes the two-dimensional space forms such as the sphere and the hyperbolic plane. Assuming that the fluid particles flow along radial geodesics, we derive the resistance functional and prove that its smooth extremals are the loxodromes of the surface. Furthermore, we analyze the constrained minimization problem, establishing the absence of strong local minima for smooth extremals, and characterizing their global minimizers.

We study the bi-Lipschitz embedding problem for a class of metric spaces called slit carpets. First we show that the $n$th stage $\mathbb{M}_n$ of the standard slit carpet of Merenkov admits a bi-Lipschitz embedding into Euclidean space with distortion $ O(\sqrt{n})$. Then, we show a nearly sharp lower bound of $Ω\left(\sqrt{\frac{n}{\log(n)}}\right)$. This result quantifies the recent result by David and Eriksson-Bique, and thus gives a quantified answer to the question 8 in the paper by Heinonen and Semmes by showing that $\mathbb{M}_\infty$ does not bi-Lipschitz embed into Euclidean spaces. Then, we study the $L^1$ embeddability of the standard slit carpet. We show that the standard slit carpet has Lipschitz dimension $1$ in the sense of Cheeger and Kleiner, and consequently prove that it admits a bi-Lipschitz embedding into $ L^1 $. Third, we generalize the results in terms of targets and domains. First, we give a qualitative and Lebesgue differentiation based argument which shows that general slit carpets do not bi-Lipschitz embed into any Banach space with the RNP property. As a consequence, $\mathbb{M}_\infty$ does not bi-Lipschitz embed to $\ell_1$. We then consider carpets $\mathbb{M}^a$ where the relative sizes of slits decrease according to a sequence $a\in c_0$. We give a quantitative $β$-number based argument which shows that the carpets $\mathbb{M}^a$ do not bi-Lipschitz embed into Hilbert space if $a\not\in \ell_{1+ε}$.

This paper studies how to balance onboard and ground computation under intermittent LEO connectivity for optimized inference freshness. As connectivity varies in time, the system switches among the actions of onboard computation, cached semantic transmission, raw-data offloading, and waiting. We define Age of Inference (AoInf) as the performance metric, where the age resets only upon successful task-valid updates. We formulate long-run average AoInf minimization as a finite-state average-cost semi-Markov decision process whose state captures the ground AoInf, orbital contact phase, cache occupancy, and cache age. We then transform the SMDP into an equivalent average-cost MDP and compute the solution via normalized relative value iteration (RVI). Numerical results indicate that the resulting hybrid policy reduces average AoInf relative to onboard-only and offload-only baselines, while requiring less computational resources on the satellite than the former, and fewer communication resources than the latter.

Multivalent binding employs multiple simultaneous supramolecular interactions, increasing avidity and selectivity compared with monovalent binding. While equilibrium aspects of multivalency are well characterized, non-equilibrium behavior remains poorly understood. By combining experiments on hyaluronic acid polymers with kinetic modeling based on stochastic chemical kinetics and molecular dynamics simulations, we systematically investigate the kinetics of multivalent binding. Notably, we find that both association and dissociation kinetics can be more selective than equilibrium binding. We explain this behavior using a two-step binding model featuring a combination of fast, weak and slow, strong interactions. These findings demonstrate a new approach: superselective targeting based on the association rate instead of the equilibrium state. The kinetic theory and experiments presented here provide a fundamental understanding of multivalent kinetics and establish design rules for superselective targeting in out-of-equilibrium systems.

Quark-gluon dynamics within protons and high-energy radiation phenomena in the universe are typically regarded as two entirely distinct fields. This paper aims to demonstrate that gluon condensation (GC) may serve as a direct bridge between these two fields. We review three key aspects of GC research: first, the Zhu-Shen-Ruan (ZSR) equation, as a nonlinear evolution equation based on structural symmetry, exhibits self-consistent connections with the DGLAP, BFKL and GLR-MQ-ZRS equations, providing a theoretical foundation for the generation of GC; second, the chaotic solutions and the shadowing-antishadowing synergy inherent in this equation can drive gluons to aggregate near the critical momentum, thereby forming a novel type of high-density, strongly interacting matter; third, these changes in microstructure manifest themselves as a broken-power-law feature in high-energy cosmic gamma-ray spectra, thereby offering new insights into the hadronic scenarios underlying certain astrophysical sources. Consequently, GC not only concerns the novel behaviour of quantum chromodynamics under extreme conditions but may also serve as a vital window for probing the deep structure of protons using cosmic-ray signals. With the advancement of higher-precision gamma-ray observations, hadron collision experiments and related theoretical research, the physical picture of GC and its observational criteria are expected to undergo more rigorous testing. Should this picture be confirmed, certain features in the high-energy gamma-ray spectrum will need to be re-examined within the deeper context of hadronic dynamics; simultaneously, GC may also provide a new entry point for research into pion condensation in nuclear physics and even condensed matter physics. Consequently, the significance of the search for GC extends beyond the model itself, reaching into multiple fields of natural science.

Modern energy systems in vehicles and built infrastructure are governed by high-dimensional dynamics spanning multiple physical domains (e.g., electrical, thermal, mechanical) and timescales. This tutorial paper presents a graph-based modeling approach created to facilitate the modeling, analysis, control, estimation, optimization, and design of these systems. Matured and validated through more than a decade of research spanning multiple academic institutions and companies, the graph-based approach combines transient energy conservation with an explicit mathematical representation of the network by which energy is stored and transferred within a system. Following a mathematical overview of graph-based models, examples of multi-domain component and system models from the recent literature are presented, including single-phase thermal systems, two-phase thermal systems, and electro-mechanical systems. This is followed by a survey of recent applications for decentralized and hierarchical model predictive control, design optimization, and control co-design. Lastly, the paper describes an open-source toolbox created to facilitate the generation and analysis of graph-based models.

In large-scale prediction problems, exhaustively following up on all test units is often impractical and inefficient, motivating a selective reporting strategy that fulfills the dual requirements of informativeness and trustworthiness. Within the InfoFCR (Informative prediction with False Coverage Rate control) framework, we propose SCIP (Selective Conformal Inference for Informative Predictions), a procedure built on three key components: (i) an informative set constructor that tailors prediction sets to individual test units according to user-specified informativeness constraints; (ii) a trust score that provides a principled quantification of the trustworthiness of candidate informative sets; and (iii) generalized conformal p-values that are used to perform FCR analysis for selecting the most promising candidates. We establish that SCIP guarantees finite-sample FCR control and is asymptotically anti-conservative, achieving higher statistical power than existing methods. The framework is highly versatile, accommodating a wide range of error metrics across both regression and classification tasks. Extensive numerical experiments on simulated and real data demonstrate the effectiveness of our approach.

A key challenge in material theory is the formulation of models that satisfy all common mechanical constitutive conditions while retaining sufficient flexibility. In this context, several important modeling aspects remain unresolved for polyconvex anisotropic hyperelasticity. We address some of these challenges and apply our results for physics-augmented neural network (PANN) constitutive modeling. The main contributions of this paper are as follows: (1) We propose a new polyconvex PANN constitutive model for anisotropic hyperelasticity based on triclinic invariants and group symmetrization. For finite symmetry groups, this model fulfills all common mechanical constitutive conditions a priori. (2) We propose a group symmetrization-based method for the construction of polyconvex invariants for finite symmetry groups. Based on this, we derive a new integrity basis for a tetragonal symmetry group and a new functional basis for a cubic symmetry group. To the best of our knowledge, these are the first polyconvex integrity or functional bases for symmetry groups characterized by structural tensors of order higher than two. (3) We provide an extensive introduction to the construction of polyconvex integrity and functional bases, which form the basis of polyconvex invariant-based constitutive models. We discuss polyconvex bases for triclinic, isotropic, transversely isotropic, monoclinic, rhombic, tetragonal, and cubic symmetry groups. (4) We benchmark the polyconvex PANN constitutive models with highly nonlinear homogenization data of cubic metamaterials.

We perform point-particle direct numerical simulations (PP-DNS) of particle-laden flow through a linear compressor cascade subjected to synthetic freestream turbulence. Monodisperse particles are advanced in a one-way coupled Eulerian-Lagrangian framework with drag-only dynamics. We quantify blade-particle collisions and resulting blade erosion based on high-fidelity data, and the erosion hotspots are predicted near the leading edge and over the pressure side. On the pressure side, for intermediate Stokes numbers, the onset of collisions correlates with elevated boundary-layer intermittency associated with bypass transition, whereas for larger particles impacts occur farther upstream with a higher probability of multiple rebounds. On the suction side, sparse collisions appear only for the smallest particles and are phase-modulated by separation-induced vortex shedding. Joint distributions of impact velocity and angle show that leading-edge impacts are faster and span wider angles than pressure-side impacts, explaining their greater erosive severity. The present results highlight the role of unsteady boundary-layer dynamics in affecting erosion in compressor cascades.

We consider a random walk on a closed manifold $M$ driven by a probability measure $μ$ on the space of $C^2$ diffeomorphisms. Provided $μ$ has compact support, satisfies certain gap and pinching conditions, and is weak-$*$ close to a volume-preserving measure, we prove that $M$ carries a unique atom-free stationary probability measure $Υ_μ$. This measure has full Frostman dimension and coincides with volume in the volume-preserving setting. Moreover, for every $x\in M$, the $n$-step distribution $μ^{*n} * δ_x$ converges to $Υ_μ$ unless $x$ is contained in a finite $μ$-invariant set. Our result applies to a variety of situations, including bi-expanding random walks on surfaces, non-linear perturbations of Zariski-dense random walks on the torus $\mathbb{T}^d$, on cocompact lattice quotients of $\mathrm{SO}(2,1)$ and $\mathrm{SO}(3,1)$, and on the sphere $\mathbb{S}^d$.