We study the question of whether the vanishing of additive invariants characterizes phantomness for smooth proper dg categories admitting geometric realizations. More precisely, let $X$ be a smooth proper variety over a field $k$, and let $\sT\subset \perfdg(X)$ be a $k$-linear admissible full dg subcategory. We construct a non-compact motive $\sM(\sT)\in \DM(k,\Q)$ and show that its $l$-adic realization recovers the $K(1,l)$-local algebraic $K$-theory of $\sT$. Analogous statements are obtained for Betti and de Rham realizations, which recover topological $K$-theory and periodic cyclic homology, respectively. As a consequence, assuming that the Chow motive of $X$ is Kimura-finite, we prove a criterion for phantomness: the vanishing of $L_{K(1,l)}K(\sT_{\overline{k}})_\Q$, of Hochschild homology in characteristic zero, or of rational topological $K$-theory over $\mathbb{C}$ implies that the rational noncommutative motive of $\sT$ vanishes. In this way, our results provide a partial answer to a question raised by Sosna. We also establish a deformation-invariance result for phantomness in smooth proper families.

We study the performance of polarizing codes over a degraded symmetric wiretap channel under a total variation distance (TVD) secrecy constraint. We show that the leakage can be bounded by the sum of the TVDs of the bit-channels corresponding to the confidential and frozen bits. In the asymptotic regime, this gives a new criterion to design wiretap codes with vanishing TVD leakage. In finite blocklength, it allows us to compute lower bounds for the secrecy rate of different families of polarizing wiretap codes over a binary erasure wiretap channel.

China's Tianwen-2 spacecraft will return samples from the near-Earth asteroid (469219) Kamo'oalewa. We previously reported that Kamo'oalewa develops an LL-chondrite-compositional, highly space-weathered surface. This study aims to estimate Kamo'oalewa's shape, regolith grain size and thickness, sub-micrometer iron (SMFe0) content, and spectral type. Using the lightcurve data and the Cellinoid model, we modeled Kamo'oalewa's shape, rotation period, and pole orientation. We then estimated its global distribution of regolith critical size using the balance method of gravity, cohesive force, and centrifugal force. Furthermore, in the temperature range of 253.15 to 473.15 K, we measured the thermal parameters of laser-irradiated LL chondrite powder that best matches Kamo'oalewa's spectrum, estimating Kamo'oalewa's thermal inertia and skin depth (lower limit of regolith thickness). Using the radiative transfer mixing model, we also estimated the content of SMFe0 in Kamo'oalewa's regolith. Finally, using the MIT online spectral classification tool for the laser-irradiated LL chondrite powder, we obtained a virtual spectral type of Kamo'oalewa. Our model gives a size of 68 m x 46 m x 39 m, a rotation period of 27.66 minutes, and a pole orientation of 134.7 degrees longitude and -11.4 degrees latitude for Kamo'oalewa. Regolith grains with a size <2 cm can remain stable over 93.8% of the global surface area of Kamo'oalewa. Laser-irradiated LL chondrite powder shows a low thermal inertia (95.5 to 135.1 J m^-2 K^-1 s^-1/2), corresponding to a thermal skin depth of 3 to 3.5 mm on Kamo'oalewa. An SMFe0 content of 0.29 +- 0.05 wt.% is required to match Kamo'oalewa's spectrum. The virtual spectral type of Kamo'oalewa is given as "Sqw".

Large Language Model (LLM)-based agentic systems have shown growing promise in tackling complex, multi-step tasks through autonomous planning, reasoning, and interaction with external environments. However, the stochastic nature of LLM generation introduces intrinsic behavioral inconsistency: the same agent may succeed in one execution but fail in another under identical inputs. Diagnosing such inconsistencies remains a major challenge for developers, as agent execution logs are often lengthy, unstructured, and difficult to compare across runs. Existing debugging and evaluation tools primarily focus on inspecting single executions, offering limited support for understanding how and why agent behaviors diverge across repeated runs. To address this challenge, we introduce InconLens, a visual analytics system designed to support interactive diagnosis of LLM-based agentic systems with a particular focus on cross-run behavioral analysis. InconLens introduces information nodes as an intermediate abstraction that captures canonical informational milestones shared across executions, enabling semantic alignment and inspection of agent reasoning trajectories across multiple runs. We demonstrate the effectiveness of InconLens through a detailed case study and further validate its usability and analytical value via expert interviews. Our results show that InconLens enables developers to more efficiently identify divergence points, uncover latent failure modes, and gain actionable insights into improving the reliability and stability of agentic systems.

In 1973 Montgomery proved, assuming the Riemann Hypothesis (RH), that asymptotically at least 2/3 of zeros of the Riemann zeta-function are simple zeros. In a previous note (arXiv:2511.20059 [math.NT]) we showed how RH can be replaced with a general estimate for a double sum over zeros, and this allows one to then obtain results on zeros that are both simple and on the critical line. Here we give a simple proof based on a direct generalization of Montgomery's proof that on assuming all the zeros are in a narrow vertical box between height $T$ and $2T$ of width $b/\log T$ and centered on the critical line, then, if $b=b(T)\to 0$ as $T\to \infty$, we have asymptotically at least 2/3 of the zeros are simple and on the critical line.

China's Tianwen-2 mission plans to return samples from a small, rapidly spinning Earth quasi-satellite (469219) Kamo'oalewa. Previous studies linked Kamo'oalewa to lunar composition and origin. Here, we propose another scenario. We reanalyzed the reflectance spectrum of Kamo'oalewa and obtained an absorption band center at 1.001+-0.028 um (error is 1sigma), consistent with LL chondrites. We then conducted space weathering (SW) experiments on meteorites and found that highly space-weathered LL chondrite powder (but not slab) successfully reproduced the reflectance spectrum of Kamo'oalewa. We further traced the dynamical origin of Kamo'oalewa and found that it probably originated from the v6 secular resonance, and more specifically, the Flora family. Kamo'oalewa exhibits a similar composition to Itokawa and 7 objects in the Flora family, but with a higher degree of space weathering. We, therefore, proposed that Kamo'oalewa probably originated from the Flora family and developed an Itokawa-compositional, highly space-weathered, fine-regolith-dominated surface.

A furthest neighbor data structure on a metric space $(V,\mathrm{dist})$ and a set $P \subseteq V$ answers the following query: given $v \in V$, output $p \in P$ maximizing $\mathrm{dist}(v,p)$; in the approximate version, it is allowed to report any $p \in P$ with $\mathrm{dist}(v,p) \geq (1-\varepsilon)\max_{p' \in P} \mathrm{dist}(v,p')$ for an accuracy parameter $\varepsilon \in (0,1)$. A particular type of approximate furthest neighbor data structure is an $\varepsilon$-coreset: a small subset $Q \subseteq P$ such that for every query $v \in V$ there is a feasible answer $p \in Q$. Our main result is that in planar metrics there always exists an $\varepsilon$-coreset for furthest neighbors of size bounded polynomially in $(1/\varepsilon)$. This improves upon an exponential bound of Bourneuf and Pilipczuk [SODA'25] and resolves an open problem of de Berg and Theocharous [SoCG'24] for the case of polygons with holes. On the technical side, we develop a connection between $\varepsilon$-coreset for furthest neighbors and an invariant of a metric space that we call an $\varepsilon$-comatching index -- a sibling of $\varepsilon$-(semi-)ladder index, a.k.a, $\varepsilon$-scatter dimension, as defined by Abbasi et al [FOCS'23]. While the $\varepsilon$-(semi-)ladder index of planar metrics admits an exponential lower bound, we show that the $\varepsilon$-comatching index of planar metrics is polynomial, all in $1/\varepsilon$. The exponential separation between $\varepsilon$-(semi-)ladder and $\varepsilon$-comatching is rather surprising, and the proof is the main technical contribution of our work.

Intrinsic instabilities of laminar premixed flames play an important role in the dynamics of hydrogen combustion and in the development of predictive models for reacting flows. However, determining their dispersion relations typically relies either on simplified analytical descriptions of the flame front or on computationally expensive direct numerical simulations (DNS). This work develops a generalized eigenvalue problem-based linear stability analysis (GEVP-LSA) framework that predicts the growth rates and spatial structure of intrinsic flame instabilities directly from the linearized governing equations of a 1D base flame. The approach is first validated using the classical Darrieus-Landau configuration, where the numerical results reproduce the analytical dispersion relation and eigenmode structure. The framework is then applied to a model flame of finite thickness governed by the reactive Navier-Stokes equations. The resulting dispersion relations and perturbation fields show excellent agreement with corresponding DNS results while reducing the computational effort by a factor of 1e8. The proposed method therefore provides an efficient and accurate tool for studying intrinsic flame instabilities and offers a scalable foundation for future stability analyses of more complex reacting-flow configurations relevant to combustion modeling and large-eddy simulations.

China plans to return samples from the near-Earth asteroid (469219) Kamo'oalewa, which we previously identified as an LL-chondrite-compositional, highly space-weathered object with fine-grained regolith. In this study, we developed 10 mL of Kamo'oalewa regolith simulant, designated "IGCAS-AST01", by irradiating LL5/6 chondrite (Kheneg Ljou^ad) powder with a high-energy pulsed laser. We then analyzed the composition, grain size distribution, density, porosity, visible to near-infrared reflectance spectrum, thermal emission spectrum, thermal diffusivity, specific heat capacity, and microstructural features of both the fresh (unirradiated) powder and IGCAS-AST01. IGCAS-AST01 is composed of 57.8 vol.% olivine, 19.9 vol.% orthopyroxene, 5.6 vol.% diopside, 12.2 vol.% plagioclase, 2.6 vol.% troilite, and minor amounts of other phases. It has a mean size of 26.99 um, a median size of 23.19 um, a density of 700 kg m^-3, and a porosity of 79.1%. Additionally, IGCAS-AST01 exhibits a low reflectance of 0.1 at 0.55 um and an extremely steep spectral slope. In the temperature range of 253.15-473.15 K, its thermal diffusivity and specific heat capacity range from 3.6-4.7 x 10^-6 m^2 s^-1 and 718.43-890.20 J kg^-1 K^-1, respectively. Furthermore, thick amorphous rims and abundant nanophase metallic iron particles are observed in olivine and pyroxene grains of IGCAS-AST01. These results could support the Tianwen-2 mission's payload calibration, sampling operations, on-orbit scientific data interpretation, and future sample analysis.

We present the first public data release (DR1) of the KMTNet Synoptic Survey of Southern Sky (KS4). This deep, wide-field imaging survey covers a southern footprint of -85$^{\circ}$ < Decl. < -28.8$^{\circ}$ in the $B$, $V$, $R$, and $I$ bands using a network of three 1.6-m telescopes. Although primarily designed to secure reference imaging for gravitational wave counterpart identification, DR1 delivers science-ready data for $\sim$4,000 deg$^{2}$ to enable a broad range of astrophysical research. The release includes deep co-added images reaching median 5$σ$ depths of 22.0-23.5 AB mag. It is accompanied by two source catalogs containing over 200 million sources with SNR $>5$: an $I$-band-selected forced-photometry catalog optimized for consistent colors, and a band-merged catalog offering enhanced completeness. Validation demonstrates robust data quality, characterized by mean astrometric offsets of $+0.054 \pm 0.129$ arcsec in RA and $-0.015 \pm 0.120$ arcsec in Dec relative to Gaia DR3. {\refbf Photometric uniformity for point sources is maintained within $\pm 0.03$ mag relative to Gaia XP for 97.5--99.8\% of the footprint across all four bands.} A key advantage of KS4 is its uniform and contiguous spatial coverage. It extends to fainter magnitudes than other uniform surveys while filling irregular gaps in existing deep datasets. All data products are publicly available via the CDS and NOIRLab's Astro Data Lab.

Recently, the Macías topology has been generalized over integral domains that are not fields, to furnish a topological proof of the infinitude of prime elements under the assumption that the set of units is finite or not open. In this article, we remove this cardinality assumption completely by using the Jacobson radical. We prove that in any semiprimitive integral domain, the group of units is not open in the Macías topology. Consequently, for a principal ideal domain, this gives an equivalence between the triviality of the Jacobson radical, the density of the set of prime elements, and the group of units not being open in the Macías topology. Furthermore, we completely characterize when Macías spaces over different infinite principal ideal domains are homeomorphic in terms of cardinalities of certain subsets of the domains. As an application we resolve an open problem concerning homeomorphism of Macías spaces over countably infinite semiprimitive principal ideal domains.

Device-independent quantum key distribution (DIQKD) provides a model of quantum key distribution with minimal assumptions and highly abstract theoretical building blocks. Although DIQKD frees us from detailed discussions of specific device models and associated error parameters, it replaces them with fundamental assumptions about the validity of quantum experiments. In this work, we propose a way to lift a protocol based on DIQKD-style assumptions to a device-dependent QKD protocol by performing local self-tests in the laboratories of the two key-generating parties. In particular, we consider routed Bell-test setups as a means of self-testing the local parties in earnest and develop a rigorous mathematical framework showing that the underlying optimization problems can indeed be transferred to the device-dependent QKD setting. As an application, we illustrate many of the relevant techniques through the case study of a routed BB84 protocol.

We study the equivariant homology of the generalized Steinberg variety of type C and show that there exists a surjective algebra homomorphism from the twisted Yangian of type $\AIII_{2n}^{(τ)}$ to it.

Cardinality estimation (CardEst) still remains a challenging problem for DBMS. Recent years have witnessed the success of ML-based cardinality estimators in outperforming traditional methods. However, these solutions suffer from poor generalizability to new data or query distribution, inability to handle complex queries, and substantial data preparation overhead, thus preventing their wide adoption in the real-world DBMS. Some recent efforts have been dedicated to addressing some but not all of these issues. We notice that the recent emerging Large Language Models (LLMs) have shown their remarkable generalizability to unseen tasks, capabilities to understand complex programs, and power to perform data-efficient fine-tuning. In light of this, we propose to leverage LLMs to mitigate the above issues. Specifically, we carefully craft prompts, and subsequently perform fine-tuning and self-correction during inference with LLMs for CardEst task. We then extensively evaluate LLMs' in-distribution and out-of-distribution generalizability, feasibility to support complex queries, and training data efficiency during fine-tuning LLMs on pre-training datasets. The results suggest that LLMs outperform the state-of-the-art in almost all settings, thus indicating their potential for the CardEst task. We further measure the end-to-end query execution time in DBMS by using the estimated cardinalities of LLMs in some practical settings, which suggests that the inference overhead of LLMs can be outweighed by the benefits brought by LLMs for CardEst.

We propose an efficient and robust protocol for the generation of entanglement between a superconducting qubit and a squeezed cavity. By applying a parametric drive to the cavity coupled to the qubit, the dynamical evolution of the system is precisely described by an anisotropic Rabi model within a squeezed reference frame. Utilizing high-order time-averaging methods, we analytically derive the resonance conditions and the effective Rabi frequency for the high-order three-photon process. By implementing an adiabatic passage, slowly tuning the cavity frequency across the resonance, the system is steered into a maximally entangled state, e.g., between the three-photon state $\ket{g,3}$ and the qubit excited state $\ket{e,0}$ in the squeezed picture. Numerical simulation results confirm the high fidelity and robustness of the proposed protocol. Our method provides a practical pathway for generating complex non-Gaussian entangled states, which are of significant value for fault-tolerant quantum computation and quantum metrology beyond the standard quantum limit.

Quantum-enhanced Markov chain Monte Carlo, a hybrid quantum-classical algorithm in which configurations are proposed by a quantum proposer and accepted or rejected by a classical algorithm, has been introduced as a possible method for robust quantum speedup. Previous work has identified competing factors that limit the algorithm's performance: the quantum dynamics should delocalize the system across a range of classical states to propose configurations beyond the reach of simple classical updates, whereas excessive delocalization produces configurations unlikely to be accepted, slowing the chain's convergence. Here, we show that controlling the degree of delocalization by adiabatically dressing the quench protocol can significantly enhance the Markov gap in paradigmatic spin-glass models.

Emerging orbitronics assumes long-range orbital current transport, analogous to spin currents. However, recent theory and experiments challenge this view, showing rather local characters for orbital polarization and orbit-spin conversions. We study angular momentum generated by ferromagnetic resonance and thermal gradients in Ni/(Pt)Ti/Au heterostructures. The observed charge current produced is independent of Ti thickness up to 60 nm, incompatible with orbital transport in Ti. Instead, its magnitude depends on both Ti interfaces, evidencing spin-mediated transport in between after and before local orbit-spin interconversions.

In this paper, we introduce the notion of a prescribed angle curve in a Riemannian manifold associated with a pair $(\mathcal{V},θ)$, where $\mathcal{V}$ is a unit vector field along the curve and $θ$ denotes the angle between $\mathcal{V}$ and the principal normal vector of the curve. When $\mathcal{V}$ is a torse-forming vector field, we establish an existence result for prescribed angle curves. In the $3$-dimensional case, we determine the curvatures of these curves in terms of the prescribed angle and the potential function of $\mathcal{V}$. Moreover, using this notion, we provide a new characterization of curves lying on geodesic spheres in real space forms.

Coherent acoustic phonons, whose damping sets the upper bound of quality factors in acoustic resonators, play a critical role in advanced telecommunication and quantum information technologies. Yet, probing their decay in the GHz regime remains challenging using conventional surface-based techniques. Dark-field X-ray microscopy (DFXM) offers a solution by enabling through-depth, non-destructive and full-field imaging of strain fields and dislocations inside bulk materials with high spatial and angular resolution. We previously used kinematic diffraction theory to describe DFXM signals based on how the Bragg peak shifts due to the strain wave, allowing us to reconstruct the frequency spectrum of coherent phonons as a function of depth through the sample. The approach of tracking the Bragg peak shifts to study phonon dynamics, however, places an upper-bound to the highest phonon frequency that can be studied, determined by the spatial resolution of the measurement. In this work, we discuss how coherent phonon dynamics can be studied with DFXM from time-dependent intensity oscillation sidebands. This approach simultaneously allows studying coherent phonon dynamics in real and reciprocal space, overcoming frequency resolution limits imposed by the real-space resolution of Bragg-peak tracking. Using Takagi-Taupin dynamical diffraction formalism, we establish the spatial and reciprocal space resolution achievable for studying the coherent phonon dynamics and evaluate conditions for observing long-lived intensity oscillations. We close by proposing experimental strategies to optimize excitation bandwidths and reciprocal-space selectivity. The formalism in the paper enables the design of DFXM experiments for quantitative, frequency-resolved measurements of acoustic phonon decay and phonon-defect interactions in bulk crystalline materials.

This work presents a novel tensor network algorithm for solving Quadratic Unconstrained Binary Optimization (QUBO) problems, Quadratic Unconstrained Discrete Optimization (QUDO) problems, and Tensor Quadratic Unconstrained Discrete Optimization (T-QUDO) problems. The proposed algorithm is based on the MeLoCoToN methodology, which solves combinatorial optimization problems by employing superposition, imaginary time evolution, and projective measurements. Additionally, two different approaches are presented to solve QUBO and QUDO problems with k-neighbors interactions in a lineal chain, one based on 4-order tensor contraction and the other based on matrix-vector multiplication, including sparse computation and a new technique called "Waterfall". Furthermore, the performance of both implementations is compared with a quadratic optimization solver to demonstrate the performance of the method, showing advantages in several problem instances.

Over the last three decades, function testing has been extensively studied over Boolean, finite fields, and discrete settings. However, to encode the real-world applications more succinctly, function testing over the reals (where the domain and range, both are reals) is of prime importance. Recently, there have been some works in the direction of testing for algebraic representations of such functions: the work by Fleming and Yoshida (ITCS 20), Arora, Kelman, and Meir (SOSA 25) on linearity testing and the work of Arora, Bhattacharyya, Fleming, Kelman, and Yoshida (SODA 23) for testing low-degree polynomials. Our work follows the same avenue, wherein we study three well-studied sparse representations of functions, over the reals, namely (i) $k$-linearity, (ii) $k$-sparse polynomials, and (iii) $k$-junta. In this setting, given approximate query access to some $f:\mathbb{R}^n \rightarrow \mathbb{R}$, we want to decide if the function satisfies some property of interest, or if it is far from all functions that satisfy the property. Here, the distance is measured in the $\ell_1$-metric, under the assumption that we are drawing samples from the Standard Gaussian distribution. We present efficient testers and $Ω(k)$ lower bounds for testing each of these three properties.

Modern software systems heavily rely on various libraries, which require understanding the API semantics in static analysis. However, summarizing API semantics remains challenging due to complex implementations or unavailable library code. This paper presents DAInfer+, a novel approach for inferring API specifications from library documentation. We employ Natural Language Processing (NLP) to interpret informal semantic information provided by the documentation, which enables us to reduce the specification inference to an optimization problem. Specifically, we investigate the effectiveness of sentence embedding models and Large Language Models (LLMs) in deriving memory operation abstractions from API descriptions. These abstractions are used to retrieve data-flow and aliasing relations to generate comprehensive API specifications. To solve the optimization problem efficiently, we propose neurosymbolic optimization, yielding precise data-flow and aliasing specifications. Our evaluation of popular Java libraries shows that zero-shot sentence embedding models outperform few-shot prompted LLMs in robustness, capturing fine-grained semantic nuances more effectively. While our initial attempts using two-stage LLM prompting yielded promising results, we found that the embedding-based approach proved superior. Specifically, these models achieve over 82% recall and 85% precision for data-flow inference and 88% recall and 79% precision for alias relations, all within seconds. These results demonstrate the practical value of DAInfer+ in library-aware static analysis.

The aim of this paper is to study the remotely almost periodic motions of dynamical systems and solutions of nonlinear differential equations. We establish some properties of remotely almost periodic motions and generalize the well known Amerio's theorem for abstract remotely almost periodic dynamical systems. Application of our general results for different classes of nonlinear differential/difference and algebraic equations is given.

Injecting world knowledge into pretrained multimodal large language models (MLLMs) is essential for domain-specific applications. Task-specific fine-tuning achieves this by tailoring MLLMs to high-quality in-domain data but encounters scalability challenges as datasets grow, necessitating a trade-off between performance and computational overhead. Existing data selection methods rely on additional scoring models or heuristic clustering, failing to concentrate on both data importance and diversity. Moreover, both methods overlook the interplay among training samples. To address these limitations, we propose CLIPPER, a training-free data selection pipeline that separates parameter and world knowledge, and leverages in-context learning to probe model responses to different demonstration-query combinations. CLIPPER identifies coresets that mirror the original dataset's perplexity distribution, preserving critical samples while maintaining diversity. Experiments on two MLLMs and three datasets show that CLIPPER matches full fine-tuning performance with significantly lower costs: Qwen2.5-VL-7B attains 47% data efficiency on VRSBench, and Llama-3.2-11B-Vision-Instruct reduces ScienceQA training time by 37%.

We extend ferromagnetic resonance (FMR) modulation theory to describe systems in which bulk and boundary states of topological materials coexist, with both appearing at the same energy. As an application of the formulation, we investigate the enhancement of the Gilbert damping constant on the $(110)$ surface of a $d$-wave superconductor where nodal quasiparticles coexist with edge states, which are one-dimensional boundary states, known as surface zero-energy Andreev bound states. We find two characteristic features: a pronounced edge-to-edge excitation peak near zero energy, and an additional edge-to-bulk excitation peak at the superconducting gap energy. We also observe power-law decay at low temperatures and exponential decay at intermediate temperatures in the low-energy regime. These features demonstrate the comparable contributions of the bulk and boundary states to the FMR response. Our theory provides a broadly applicable framework for the analysis of topological materials.

In this paper, we study a class of one-dimensional nonlocal nonlinear Schrödinger equations on the line with nonlinearity given by a Fourier multiplier whose symbol has subcritical high-frequency growth. In terms of symbol order, this class is intermediate between the cubic nonlinear Schrödinger equation and the Calogero--Moser derivative nonlinear Schrdöinger equation. We prove local well-posedness in $L^2(\mathbb{R})$ throughout the full scaling-subcritical range. Due to derivative loss, the standard Duhamel integral is not directly meaningful for rough data. To avoid this problem, we first construct the propagator $S_V$ for rough time-dependent potentials $V$, and then prove an Ozawa-Tsutsumi type bilinear Strichartz estimate for the perturbed flow $S_V$. These linear theories yield a concrete construction of rough solutions without using any equation-specific algebraic structure. For real-valued symbols, mass is conserved, and the local solutions are therefore global.

This paper investigates domain hemivariational inequality problems arising from the non-stationary two- and three-dimensional convective Brinkman-Forchheimer extended Darcy (CBFeD) equations, which describe the flow of viscous incompressible fluids through saturated porous media in bounded domains. These equations may be regarded as generalized Navier-Stokes systems incorporating both damping and pumping mechanisms. For all admissible absorption exponents $r \ge 1 $ and effective viscosity $μ> 0 $, the existence of weak solutions to the non-stationary 2D and 3D CBFeD equations with hemivariational inequalities is established via a regularized Galerkin approximation scheme, based on a suitable regularization of the Clarke subdifferential. A noteworthy aspect of the analysis is that the existence results extend to the three-dimensional non-stationary Navier-Stokes equations. Moreover, under appropriate conditions on the absorption exponent, specifically, $r \ge 1 $ in two dimensions and $ r \ge 3 $ in three dimensions, it is shown that weak solutions satisfy the energy equality. In addition, uniqueness of solutions is proved for $ r \ge 1$ in 2D and $r \ge 3$ in 3D, with the additional requirement $2βμ> 1 $ in the critical case $r = 3 $.

Telek (2022) asked whether a rational arrival process (RAP), specified by matrices ${G}_0$ and ${G}_1$ and an initial row vector $ν$, with strictly positive joint densities and a unique dominant real eigenvalue of ${G}_0$ must admit an equivalent Markovian arrival process (MAP). A counterexample of order $3$ is given, showing the answer is no, and that the conjecture fails even under the stronger condition of exact normalisation $({G}_0+{G}_1){1}={0}$. The construction combines a strictly positive exponential baseline with a two-dimensional correction driven by an irrational rotation. Strict positivity of all joint densities follows from the continuous-time damping of the correction block; the obstruction to MAP realisability comes from the poles of the boundary generating function at $e^{\pm iφ}$, which cannot be peripheral eigenvalues of any finite nonnegative matrix when $φ/π$ is irrational.

We report the results of $g-$, $r-$, and $i-$dropout selections based on optical identifications of Very Large Array Sky Survey (VLASS) radio sources using the Hyper Suprime-Cam Subaru Strategic Program survey (HSC--SSP). By positional crossmatching within $1''.5$ between the VLASS Epoch~2 catalog and the HSC--SSP Wide-layer catalog ($i \lesssim 26$), we obtain $\sim$400 high-redshift radio AGN candidates at $z \gtrsim 4$ over a $\approx1200~\mathrm{deg}^2$ survey footprint, extending optimistically to $z \sim 7$. Optical magnitudes cluster at $i_\mathrm{AB} \simeq 24$--26, indicating that these sources are largely inaccessible to shallower surveys such as SDSS. By further cross-matching the HSC--VLASS dropout catalog with VLA Faint Images of the Radio Sky at Twenty-centimeters (FIRST) at 1.4~GHz, the LOFAR Two-metre Sky Survey (LoTSS) at 144~MHz, and the TIFR GMRT Sky Survey (TGSS) at 150~MHz, the majority of the high-$z$ candidates show flat to moderately steep radio spectra ($-1 \lesssim α\lesssim 0$, with $f_ν\propto ν^α$), and some also exhibit turnover radio spectra, demonstrating that conventional ultra-steep-spectrum (USS; $α<-1.3$) selection would miss most of the population selected in this study. Building on this, we perform SED fitting and obtain AGN luminosities, which show a clustering at typical bolometric luminosities of $\log(L_{\rm bol}/{\rm erg~s^{-1}})\sim46$--47. We also examine the comoving number density distribution of our samples and find a sharp decline around the $i$-dropout regime ($z \sim 6$), suggesting the possible disappearance of luminous radio AGNs toward the epoch of reionization.

Illicit online promotion is a persistent threat that evolves to evade detection. Existing moderation systems remain tethered to platform-specific supervision and static taxonomies, a reactive paradigm that struggles to generalize across domains or uncover novel threats. This paper presents a systematic study of In-Context Learning (ICL) as a unified framework for illicit promotion detection. Through rigorous analysis, we show that properly configured ICL achieves performance comparable to fine-tuned models using 22x fewer labeled examples. We demonstrate three key capabilities: (1) Generalization to unseen threats: ICL generalizes to new illicit categories without category-specific demonstrations, with a performance drop of less than 6% for most evaluated categories. (2) Autonomous discovery: A novel two-stage pipeline distills 2,900 free-form labels into coherent taxonomies, surfacing eight previously undocumented illicit categories such as usury and illegal immigration. (3) Cross-platform generalization: Deployed on 200,000 real-world samples from search engines and Twitter without adaptation, ICL achieves 92.6% accuracy. Furthermore, 61.8% of its uniquely flagged samples correspond to borderline or obfuscated content missed by existing detectors. Our findings position ICL as a new paradigm for content moderation, combining the precision of specialized classifiers with cross-platform generalization and autonomous threat discovery. By shifting to inference-time reasoning, ICL offers a path toward proactively adaptive moderation systems.