Papers

We proof the existence of the Integrated Density of States (IDS) for fractional random Schrödinger operators with Gaussian potential based on the theory of isotropic $\alpha$-stable Lévy processes. Further we show that the IDS exhibits Lifshitz tails and determine its asymptotics at the right end of the spectrum. Isotropic$\alpha$-stable Lévy processes are self-similar but not continuous. Thus self-similarity is a fundamental concept for these quantum operators.

Reliable and total-order broadcasts in the Byzantine-fault model are well studied, but adding causal order has received comparatively little attention, largely due to the complexity that stems from actions of Byzantine processes. Existing solutions almost exclusively build causal ordering on top of total-order broadcast. The combination of causal order with reliable broadcast remains rare, and the few solutions that exist adopt the classical definition of causality based on events occurring at individual processes (the happened-before relation). This definition is not sufficient to enforce causal ordering among broadcast messages: Byzantine processes can lie about, omit, and forge dependency information and thereby violate the causal order among self-reported events. Such manipulations remain indistinguishable from correct behavior to any single observer. We demonstrate the issue and its consequences via a front-running attack. To close this gap, we extend the notion of reliable broadcast to externalize local potential knowledge. We use this to formalize the first complete definition of causal message ordering in reliable broadcast under Byzantine faults. Unlike the classical formalization, this notion is grounded in the joint observations of a sufficiently large group of correct processes rather than a process's own view. Building on this definition, we characterize the properties of a Byzantine reliable broadcast channel that guarantees causal ordering. We then present an efficient protocol that satisfies these properties: it is resilient to the optimal number of $f < n/3$ Byzantine faults and for one instance that broadcasts payload message~$m$, it has bit complexity $O(n^2(|m| + \lambda + n))$, where $\lambda$ denotes the maximal size of a unique (cryptographic) label for~$m$. Finally, we prove the protocol achieves Byzantine reliable broadcast with causal ordering.

We study the problem of minimizing a weighted sum of squared Wasserstein distances with signed coefficients. In the case of a single positive coefficient, we derive two convex dual formulations: one expressed in terms of Brenier potentials, and one as a projection problem in convex order. Such dual formulations arise via Toland duality by exploiting different notions of convexity within the problem. They generalize those recently established for the metric extrapolation problem, which corresponds to signed barycenters between only two measures. For general signed barycenters, we identify necessary optimality conditions that reduce the problem to the case with one positive coefficient. On the numerical side, we propose a grid-free, particle-based algorithm built on entropic regularization and Sinkhorn iterations to compute signed barycenters with arbitrary coefficients.

We classify finite-energy positive solutions to the Yamabe-type equation on the 15-dimensional octonionic Heisenberg group, whose algebra is non-associative. Extremals for the Folland-Stein-Sobolev inequality on this group are explicitly described. This confirms the conjecture of Garofalo and Vassilev [Duke Math. J. 2001] on the $15$ dimensional octonionic Heisenberg group.

This supplementary material provides the complete mathematical development for the signal-to-noise ratio loss (SNRL) analysis of pulse-wise spatially randomized encrypted sensing. We first derive a reduced model in which the transmit-side randomization is represented by an independent and identically distributed complex spatial coefficient. Based on this model, we establish the almost-sure and mean asymptotic SNRL, derive the leading bias and variance for a finite coherent processing interval (CPI), and obtain closed-form coefficient moments and asymptotic SNRL for independent Bernoulli element activation. These results complement the concise statements and proof sketches in the associated paper.

Let $\Delta_k$ be the Dunkl Laplacian associated with an arbitrary root system and a nonnegative multiplicity function. For every $0<p\leq1$, a Coifman-Weiss type atomic characterization of the Hardy space $H^p_{\mathrm{Dunkl}}$ is established. More precisely, it is proved that the space $H^p_{\mathrm{Dunkl}}$, which is originally defined by a relevant square function, coincides with the space generated by $({\rm CW},p,2)$-atoms, that is, atoms supported on Euclidean balls and satisfying cancellation conditions against all polynomials of degree $ \leq s_p$, where \[ s_p=\left\lfloor \mathbf N\left(\frac1p-1\right)\right\rfloor \] and $\mathbf N$ is the homogeneous dimension of the underlying Dunkl measure. The corresponding quasi-norms are equivalent. We also show that the same space is obtained when the $L^2$ size condition in the definition of atoms is replaced by the $L^\infty$ size condition. The strategy of the proof is to use an operator-type atomic decomposition associated with the Dunkl Laplacian, and then prove that each such operator atom can be written a linear combination of $({\rm CW},p,2)$-atoms.

Recent studies of the superconducting proximity effect in quasicrystalline and topological systems have opened up a new research direction for exploring how quasiperiodicity and topology influence proximity induced superconductivity. In this work, we investigate spatial variation of the proximity induced pairing amplitude in a hybrid ring composed of a spin singlet superconductor and a normal region described by three different lattice models using the self-consistent Bogoliubov-de Gennes formalism. We first consider the normal region described by the diagonal Aubry-André-Harper (AAH) model. Increasing the quasiperiodic potential enhances spatial fluctuations in the induced order parameter, while progressively suppressing its magnitude in the normal region. Beyond the localization transition, proximity-induced pairing is strongly diminished due to the localized nature of the underlying electronic states. The normal region is then modeled by the Su-Schrieffer-Heeger (SSH) chain to investigate the effect of hopping dimerization. Weak dimerization introduces oscillatory modulations in the induced pairing that extend deep into the normal region, whereas strong dimerization confines these oscillations and significantly reduces the penetration of superconducting correlations. Finally, we study the combined SSH-AAH model to explore the interplay between the dimerized hopping and quasiperiodicity. The results show that the SSH dimerization determines the oscillatory behavior and penetration of the induced pairing, while the AAH potential enhances spatial inhomogeneity and further reduces its magnitude. Together, these two effects provide a versatile means of controlling proximity-induced superconductivity in quasiperiodic hybrid systems.

Congenital heart disease (CHD) diagnosis and surgical planning often require patient-specific 3D anatomical models, but manual segmentation is labor-intensive, particularly in complex anatomies. Although deep-learning methods can automate this process, they are typically evaluated in-distribution, despite clinically relevant shifts in scanner, protocol, institution, population, and imaging modality. We present, to our knowledge, the first systematic evaluation of out-of-distribution (OOD) generalization in CHD segmentation, using ImageCHD as a held-out target cohort. We compare representative segmentation architectures under combined CT and CMR training, CT-only training, self-supervised pretraining, and limited target-domain adaptation. In-distribution performance proves to be a poor indicator of cross-cohort robustness: nnU-Net achieves the highest validation Dice (0.77) but falls to 0.51 on ImageCHD, while SwinUNETR generalizes substantially better, reaching 0.67 Dice. MAE and JEPA pretraining provide only modest additional benefit, suggesting that architecture contributes more to robustness than the tested pretraining strategies in this setting. When limited target-domain supervision is introduced, all SwinUNETR variants exceed 0.76 Dice with only 11 labeled ImageCHD cases. These findings demonstrate that conventional in-distribution evaluation can obscure clinically important generalization failures and support explicit cross-dataset testing as a key component of CHD segmentation evaluation.

Indirectly encoded neural networks can assign different activation functions to individual nodes, but the right functions are rarely known in advance. When the available set contains only standard monotonic functions, problems like parity become unsolvable, yet an all-inclusive palette underperforms a curated one. How should evolution discover which functions to use? We address this as a meta-learning problem, designing 13 strategies (11 inspired by biological adaptation mechanisms, plus baseline and oracle controls) that modify the set of available activation functions during evolution. Each strategy translates a biological principle into an evolutionary operator: for example, circadian-inspired oscillatory gating cycles functions in and out of the palette on a fixed schedule, while immune-inspired Clonal Selection permanently protects functions that consistently correlate with fitness. We evaluate all strategies across more than 3,000 runs on parity and non-parity problems, first evolving the activation palette alone, then co-evolving a per-node aggregation palette on harder problems; an independent replication with new seeds confirms a stable high-reliability tier, with Circadian holding its top rank. Bio-inspired strategies match the solve rate of a tuned baseline but converge up to twice as fast, with Circadian halving total compute. Strategy rankings reverse across problem types, with no strategy dominating all domains. Strategy success is largely shaped by timescale compatibility: strategies whose characteristic timescale matches the evolutionary evaluation window consistently outperform those that operate too slowly. The practical guideline: match the mechanism's timescale to the evaluation budget. Rescaling the slowest strategy bypasses the oscillatory barrier entirely: all nine solutions solve parity with non-oscillatory activations paired with min or max aggregation.

We study the far-field behavior of solutions of the one-phase Bernoulli free boundary problem in the exterior of a ball, and of entire solutions with a single compactly supported inhomogeneity of the free boundary condition, which we call a defect. For solutions which blow down to a half-plane solution (proper solutions) we establish an asymptotic expansion at infinity: in dimension $d \geq 3$ the free boundary height converges to a limit at rate $|x|^{2-d}$ with a capacity-type coefficient, while in dimension $d=2$ the expansion carries a logarithmic term. A significant novelty is that the expansions are quantitative and uniform over all the proper solutions.

AI assistance places a metacognitive demand on users, who must judge their own competence and the system's. Yet designers lack comparative evidence on which interventions to choose, where to place them, and how to tell whether they worked. We elicited 30 interventions from 11 experts and, with prior work, organized them into a design space of time (when an intervention acts), level (whose competence is judged), and source (who supplies the monitoring cue). A between-subjects experiment (N = 917; 12 planning-and-organizing problems) compared a per-task reliability card, contrasting replies, pause points, and post-problem reflection against a baseline LLM assistant. Reliability cards and contrasting replies reduced estimation error and overconfidence and increased aggregate confidence discrimination. No task-performance improvement or average within-item discrimination gain was established. We contribute a shared vocabulary, a design space, and evidence that measured monitoring and task performance are separable design targets.

Assistive autonomous systems must anticipate human goals before an observed behavior is complete. This article formulates anticipation as goal inference from a partially observed multimodal episode together with structured prediction of the remaining behavior, rather than exact motor forecasting. A compact Hierarchical Planning Decoder (HPD) is attached to a frozen neuro-symbolic recognition encoder and predicts, at four ontological levels, the next actions, the remaining activities and low-level intentions, and the episode high-level intention(HLI). The decoder is trained with soft neuro-symbolic regularization combining transition-coherence and hierarchical continuity losses, and is decoded with hard reachability masks that enforce ontological validity at inference. On a compositional four-level benchmark of 15,002 multimodal episodes built over NTU RGB+D 120 features, three headline properties are observed together. The advantage over the strongest sequential baseline grows with the anticipation horizon, from +1.7 points at step 1 to +7.3 points at step 3 (top-5). Under compositional generalization, where one parent association per multi-parent low level intention is held out, this advantage widens to +4.9 points at step 1. At the episode level, 96.8% of anticipated trajectories satisfy the joint logic constraints, above the 88.1% strongest-baseline value and the 73.9% ground-truth floor; soft logic terms alone account for a 59.8 to 71.1% relative reduction of HLI-reachability violations, and the hard masks then eliminate them entirely. Neural generation supplies predictive ranking, symbolic constraints supply onto logical validity, and their combination yields coherent hierarchical anticipation while exposing remaining challenges in compositional goal generalization and unordered set prediction.

Muon-spin rotation/relaxation is one of the most powerful microscopic probes for understanding magnetic order, spin dynamics, superconductivity, and complex magnetic phases in emerging quantum materials. Quantitative interpretation depends critically on the accurate identification of the muon stopping site, a problem traditionally addressed using density functional theory-based structural relaxation. However, conventional DFT approaches are computationally demanding, time-consuming, and require extensive calculations. Here, we develop MEOWN (Muon Engine for Optimized Weighted Networks), a machine-learning-based, physics-informed computational framework that combines a Polarizable Unperturbed Electrostatic Potential (P-UEP) model with machine-learning-guided optimization and symmetry-driven relaxation to predict energetically favorable muon stopping sites in crystalline solids. MEOWN explicitly incorporates electrostatic interactions, electronic screening, polarization effects, and zero-point motion into the learning workflow, providing physically interpretable predictions with substantially reduced computational cost while maintaining physical consistency and scientific rigor. To validate the framework, we investigated several well-established benchmark materials, including MnSi, CoF2, CaF2, LiF, and NaF. The predicted muon stopping sites and dipolar fields are in close agreement with previously reported DFT+mu results, demonstrating the reliability and transferability of the approach across chemically diverse systems. Notably, MEOWN predicts the equilibrium muon stopping site within a few minutes, offering a significant speedup over conventional DFT-based calculations. This manuscript presents the theoretical foundations, computational methodology, and validation of MEOWN as a general-purpose software for rapid and reliable muon stopping-site prediction in crystalline materials.

Asset Administration Shells (AAS) provide a standardized means of representing assets and their information in manufacturing and increasingly serve as a basis for software services. However, different AAS instances vary in structure, content, and degree of completion, making it difficult to determine whether a given AAS is suitable for a specific application. This paper presents two complementary methods to support the comparison and application-oriented assessment of AAS. First, set-theoretic operations are employed to compare AAS models, enabling the identification of common, missing, and differing submodels and parameters. Second, an AAS suitability model assesses the conformity of an AAS to the requirements of a specific use case. The assessment considers structural conformity, semantic consistency, cardinality, and specification conformity and can be performed either against a reference AAS or a set of required SemanticIDs. A suitability value is derived from the identified deviations and is complemented by a detailed report of missing or non-conforming information. The proposed approach support practitioners and researchers in the comparison of evolving AAS and provide application-specific information on their suitability for manufacturing software services.

Message-passing Graph Neural Networks (GNNs) iteratively propagate and aggregate local neighborhood information followed by global readout to learn graph representations. However, their discriminative power is upper-bounded by the Weisfeiler–Lehman (1-WL) graph isomorphism test. This prevents GNNs from distinguishing certain non-isomorphic graphs with identical local neighborhood structures, often leading to similar graph representations. Unified Topological Signatures (UTS) capture compact, multi-scale representation of global graph topology derived from persistent homology. We introduce two complementary UTS signatures: Graph_UTS- a static signature of the input graph topology, and Embedding_UTS- a dynamic signature of the evolving embedding topology. They encode structural information inaccessible to 1-WL-based message-passing GNNs, yet their capabilities are explored solely for post-hoc embedding-space analysis. We integrate UTS into GNN training across three architectural interventions: (i) UTS-Aug: augmenting with standard readout feature that encodes graph's true topology; (ii) UTS-Reg: topological regularizer that constrains representation collapse; (iii) UTS-Pool: topology-guided pooling that retains structurally critical nodes. We further leverage UTS as a layer-wise diagnostic to quantify oversmoothing during GNN training. Theoretically, we show that integrating UTS into GNN optimization strictly extends GNN expressivity beyond the 1-WL hierarchy. Experiments on three graph classification benchmarks show consistent benefits: Graph-UTS, Dual-UTS, and UTS-Pool improve accuracy across all three datasets, Embedding-UTS provides smaller but similarly consistent gains, and UTS-Reg's benefit varies across graph domains. Accuracy improves by up to 5.8% with Graph-UTS augmentation, by up to 1.9% with UTS-Reg, and achieves comparable performance to TOGL with UTS-Pool.

We develop an extension of Deligne–Lusztig theory to certain (possibly infinite type) disconnected reductive groups arising from the special fibers of point stabilizers in the Bruhat–Tits building, which we call paraductive. We then compute explicit lower bounds for the Tate cohomology of representations of paraductive groups, relating these to Shintani descent, Lusztig restriction, and the Glauberman correspondence. As an application, using Feng's modular functoriality and Scholze's independence of $\ell$, we compute the Fargues–Scholze L-parameters of non-singular depth $0$ cuspidal representations of a (possibly wildly ramified) reductive group over a nonarchimedean local field.

DoA estimation often requires not only accurate recovery of source directions, but also reliable characterization of the uncertainty in these estimates. While classical subspace methods such as ESPRIT provide principled uncertainty analyses, their performance and uncertainty quantification rely on restrictive assumptions. Recent deep learning approaches alleviate these limitations and enable robust. DoA estimation in challenging conditions, but generally provide point estimates and lack principled uncertainty characterization. In this work, we develop an AI-aided framework for joint DoA estimation and uncertainty quantification that combines the robustness of model-based deep learning with the analytical foundations of classical subspace methods. Building on AI surrogate covariance recovery, we extend existing ESPRIT uncertainty analyses to characterize the full covariance structure of the DoA estimation error and integrate this characterization into a subspace-oriented deep learning architecture. We further propose a dedicated learning strategy that jointly promotes accurate DoA recovery and faithful uncertainty estimation. The resulting methodology preserves the interpretable processing pipeline of classical subspace methods while enabling reliable operation in regimes where conventional approaches struggle. Our numerical studies demonstrate that the proposed framework consistently achieves accurate DoA estimation together with reliable uncertainty characterization across diverse challenging scenarios, including coherent sources, limited snapshots, and array calibration errors.

Linearly constrained minimum variance (LCMV) beamforming is a fundamental technique for controlling the array response toward multiple desired signals while suppressing interference and noise. In practical systems, however, the steering vectors involved in the response constraints are often affected by direction-of-arrival errors, calibration imperfections, and other array uncertainties. Such mismatches may cause constraint violations and substantial performance degradation. This paper investigates multi-signal LCMV beamforming under steering-vector mismatch and develops probabilistic constraints to guarantee a prescribed reliability level for each desired signal. Several uncertainty-information settings are considered. When the exact mismatch distribution is known and belongs to the class of complex elliptically symmetric distributions (CES), a distribution-informed convex safe reformulation is derived using exact CES quantiles. When the mismatch distribution is unknown, and only partial statistical information or empirical data is available, robust probabilistic reformulations are developed under moment-based ambiguity, support-based uncertainty, and Wasserstein data-driven ambiguity, and the resulting deterministic formulations can be expressed as convex second-order cone programs. Numerical experiments show that the proposed designs improve out-of-sample reliability and reduce beam-gain sensitivity to steering-vector mismatch, while maintaining competitive output signal-to-interference-plus-noise (SINR) ratio and robust beampatterns.

We present Budgeted Express-Mesh, a topology-routing co-design that adds a small number of traffic-aware express links under a fixed wire budget. An ASPL-based greedy placement is refined by simulation-guided annealing, while packets use committed top-K routes selected from delayed express-link congestion and reservation signals. Across four synthetic workloads, optimized placements consistently improve high-load throughput over Mesh and random placement, and annealing further improves Greedy. The gains persist under delayed quantized congestion information, longer express-link latency, multi-flit packets, and a 16-by-16 heterogeneous workload.

Current predictive turn-taking models (PTTMs) achieve strong performance on benchmarks with controlled acoustic conditions and clean audio signals. Their generalisation to conversations with overlapping speech and background interference remains underexplored. In this research, we evaluate audio-visual PTTMs trained with clean data on a challenging cocktail-party testbed derived from the AVCocktail dataset, and analyse their adaptation behaviour to this new domain. Experimental results show consistent performance degradation across audio and visual modalities under noisy conditions, with up to 38% relative drop in weighted F1. Fine-tuning on the new domain improves robustness, but gains vary across modalities and depend on the size of the available pre-training data. These findings provide insights into the different generalisation and adaptation capabilities of the audio and visual modalities, and indicate the need for robust modelling strategies to adapt to the complexities of human interactions in noise. All code and turn labels are made publicly available to facilitate further research.

We establish a weighted extension of the Bishop-Gromov volume comparison theorem for complete Riemannian manifolds with Ricci curvature bounded below. Given a point $p$ and positive radial weights $f$ and $h$, we consider the weighted volume of a geodesic ball and a corresponding model weighted volume in the simply connected space form of constant sectional curvature $k$. We prove that, under the assumption that $h/f$ is nondecreasing, the ratio of these two weighted volumes is nonincreasing with respect to the radius. We also characterize the equality case, showing that equality forces the weight ratio to be constant and the corresponding geodesic ball to be isometric to the model ball. As applications, we derive weighted Bishop-Gromov-type comparisons, annular comparison inequalities, and weighted volume doubling estimates. We further apply the monotonicity formula to obtain weighted area-volume inequalities and sharp weighted isoperimetric-type comparisons for geodesic balls.

Let $U,V$ be matrix $\mathcal A_p$ weights, $1<p<\infty$, and let $B$ be a matrix-valued function. For Calderón–Zygmund operators $T$ with scalar standard kernels, we characterize boundedness and compactness of $[M_B,T\otimes I_m]:L^p(U)\to L^p(V)$ by two-matrix BMO and vanishing oscillation conditions. The converse statements require non-degeneracy condition. For matrix \(\mathcal A_2\) weights \(U,V\) and a fixed Riesz transform \(R_j\) on \(\mathbb R^d\), \(d\geq2\), we characterize the Schatten class membership of \(C_{B,R_j}:L^2(U)\to L^2(V)\). We characterize $\mathcal S^r$ membership for $r>d$ and $\mathcal S^{d,\infty}$ membership by $\ell^r$ and weak $\ell^d$ conditions on the two Hilbert–Schmidt cube oscillations. For $0<r\leq d$, $\mathcal S^r$ membership is equivalent to $B$ being constant almost everywhere. The weak endpoint condition implies a weighted first-order Sobolev estimate for $C^1$ symbols, with a converse for constant or uniformly elliptic weights.

We introduce forward recursive aggregator systems in Itô-diffusion markets, bridging the theories of recursive utilities and of forward performance criteria. The resulting criteria are constructed forward in time and accommodate arbitrary or rolling horizons. We derive the associated HJB SPDE and sufficient conditions for consistency and admissibility. We also develop a convex duality theory based on state price densities and identify the density generated by the optimal wealth process. For homothetic aggregators, we characterize a class of forward Epstein–Zin recursive utility processes, derive the optimal investment and consumption policies, and obtain explicit solutions in the Black and Scholes and Heston models.

We prove a stronger form of Cowling's $L^p$-integrability conjecture for an admissible class of Kunze–Stein groups containing all $S$-algebraic groups and all closed CCR boundary-transitive subgroups of tree automorphism groups. We also disprove a cyclic version formulated by Samei–Wiersma.

Academic collaboration is of self-evident value but requires a quantitative representation to be optimally guided by policy. No established methodological approach to such representation exists. Here we introduce a general framework of graphical and bibliometric analysis of open data for the task of quantifying the impact of academic networks, with NIHR Biomedical Research Centre (BRC) clinical-academic partnerships in England as the prototype. We define publication-level identities for the 20 English BRCs based on the conjunction of authors from each BRC's partner institutions. Drawing on bibliometric and administrative records, we characterise the graphical properties of each network, estimate what the university adds to the hospital's papers, what the partnership adds to the papers of relatively infrastructure-poor collaborating institutions, and how that gain depends on existing infrastructure. We apply our framework to the NIHR UCLH/UCL BRC as an exemplar. UCLH/UCL authored 20,985 network papers from April 2007, in collaboration with 9,868 distinct external partners over the whole record, forming the most central node of the graph of networks across England. University co-authored papers exhibited 1.6 times the field-weighted citation impact (FWCI) of hospital-only papers, and were 2.1 times as likely to be cited by a patent. Across 60 of the exemplar's most partnered with UK healthcare organisations, the benefit rose from 1.8 times where local NIHR infrastructure activity was densest to 3.4 times where it was sparsest, while impact without the exemplar varied little. Academic networks can be robustly identified from open data, enabling comparative analysis of collaborative impact. Applied to NIHR BRCs, the approach enables quantification of the impact across networks and reveals that benefit is most pronounced where infrastructure is least developed.

We prove that a general smooth complete intersection of positive dimension and codimension at least 2 over an algebraically closed field has trivial linear automorphism group, except in the case of two quadrics. The proof uses incidence varieties and treats separately prime-to-characteristic and unipotent automorphisms.

Multi-state HLL-type approximate Riemann solutions of ideal magnetohydrodynamics (MHD) typically assume that the medium within the Riemann fan is incompressible, and thus the slow magnetoacoustic mode cannot be included in their space-time wave configurations for the intermediate states. We propose a new strategy to design multi-state HLL-type solutions, allowing for the medium within the Riemann fan to be compressible. In particular, for the complete seven-wave configuration of the MHD Riemann problem, we estimate the slow magnetoacoustic speeds and a longitudinal flow speed between the slow modes, whose compressibility is estimated based on the relations between fast, Alfvén, and slow speeds. Then, following the conservation laws and Rankine-Hugoniot jump relations across the fast, Alfvén, and slow waves, we calculate all the intermediate states within the Riemann fan. Moreover, we discuss the solutions when certain wave modes degenerate, ensuring well-posedness and smooth transitions between complete and degenerate wave configurations. Numerical simulations using a Finite Volume solver show that the proposed approximate Riemann solution is less diffusive than the classic HLLD scheme, particularly for slow mode waves. For example, in a 1D test case with a strong longitudinal magnetic field, the new scheme needs one order of magnitude fewer grid cells compared to the classic HLLD scheme to resolve all wave modes.

We study nonconvex methods for matrix completion, the problem of recovering a low-rank matrix from a subset of its entries. Convex methods achieve sample complexity linear in the matrix dimension and the rank, up to logarithmic factors, whereas global guarantees for commonly used nonconvex methods require a higher polynomial dependence on the rank. We close this gap by analyzing Riemannian gradient descent (RGD) and Riemannian Gauss–Newton (RGN) methods. For an $n\times n$ matrix of rank $r$ with incoherence parameter $\mu$ and condition number $\kappa$, the two methods achieve exact recovery with high probability from $O(\mu nr\log n\log(n\kappa))$ and $O(\mu nr\log n\log(2\mu r\kappa))$ observations, respectively. The methods use a multiscale residual initialization, while the analysis simultaneously controls the spectral error and incoherence. The resulting RGD iterates converge linearly, whereas RGN eventually converges Q-quadratically.

In active media, a moving passive object can experience negative drag and receive positive mechanical power from the bath, in contrast to the dissipative response in equilibrium. For a probe moving at velocity $\mathbf{V}$ and interacting weakly with an ideal active bath, we show that free-particle trajectory statistics determine the density response and the resulting bath force. The probe–particle interaction sets the wave-vector weighting, while the imposed motion selects frequencies $\omega_{\mathbf q}=\mathbf q\cdot \mathbf V$, separating bath dynamics from probe properties. The sign of the response spectrum yields probe-independent constraints for mechanical power transfer. Applying this framework to representative active-particle models reveals distinct power-transfer behavior despite identical displacement variances. For prescribed probe motion, the relation predicts drag reversal, motion-induced density distortions, and velocity-dependent friction, in quantitative agreement with simulations. Beyond the mean force, the free-particle trajectory statistics also determine the force autocorrelation. These results provide a route to predict the mechanical response of probes from single-particle trajectory statistics measured in the absence of the probe.

Disk fractions in star-forming clusters decline with stellar age, and are routinely read as the lifetime distribution of primordial protoplanetary disks. We argue that this reading is not warranted, because late infall can replenish or reform disks after collapse. To formalize this, we model the disk fraction as a superposition of primordial and environmentally replenished components, with region-to-region dispersion in the supply rate. Disk fractions cannot distinguish a long-lived disk from a short-lived one that is repeatedly replenished. In Sco–Cen both reproduce the data, the replenished model marginally better. Stellar age therefore need not equal disk age, and stars can host successive disk generations with distinct mass, orientation, and chemistry. A key test is whether regions of the same stellar age show a genuine spread in disk fractions, as environmental variation in the replenishment supply predicts.