Papers

Backdoor repair aims to suppress malicious behavior in compromised models while preserving benign task performance. Existing studies typically evaluate these objectives using Attack Success Rate (ASR) and Overall Clean Accuracy, but aggregate clean accuracy can obscure substantial degradation concentrated in a small portion of the label space. We revisit benign-performance evaluation from a preservation perspective by distinguishing aggregate clean utility from the preservation of previously available class-wise performance. We define class-wise preservation loss by comparing clean performance before and after repair and show that aggregation can hide localized degradation through localized-loss dilution and cross-class compensation. To complement Overall Clean Accuracy, we characterize localized preservation loss using Worst-Class Preservation Loss and Tail Preservation Loss. We conduct a systematic empirical study across representative backdoor attacks, repair methods, datasets, attack targets, and model architectures, with additional validation under clean-label attacks. Results show that effective attack suppression and favorable aggregate clean performance do not necessarily imply uniform preservation of previously available benign performance across classes. Substantial localized preservation losses can remain, and their severity and class-wise structure vary across repair conditions. These findings motivate preservation-oriented class-wise evaluation alongside ASR and Overall Clean Accuracy.

Urban-scale 3D reconstruction from satellite imagery supports disaster response, city monitoring, and geospatial digital twins, yet neural rendering methods typically optimize average visual fidelity rather than the structures that analysts inspect first: buildings. We present an agentic building-aware satellite Gaussian Splatting workflow that uses Segment Anything-derived building masks as semantic priors and an Agentic Reconstruction Controller to select, verify, and record DSM reconstruction policies. On the DFC2019 JAX\_004 scene, building-aware weighting reduces building-region DSM MAE from 0.844 m to 0.806 m, showing that semantic priors can shift reconstruction capacity toward analyst-critical regions. A staged schedule provides a balanced operating point, improving full-scene MAE from 1.362 m to 1.349 m while retaining a building gain. Across four JAX scenes, the Agent selects validated policies for both general DSM and building-focused DSM objectives, and produces building-inventory metadata and per-scene decision records. The system combines semantic priors, policy selection, region-specific DSM metrics, and DSM-derived GIS surface products for auditable urban 3D analysis.

Most minimally invasive procedures are still performed with hand-held laparoscopic instruments, yet only the endoscopic video is retained; the instrument motion that expresses surgical skill, and that could support skill assessment and robot learning, is lost. Pose from video alone remains millimeters to centimeters off, and an instrument-mounted inertial measurement unit (IMU) cannot rely on its magnetometer, whose field changed with tool pose and between sessions in our measurements. We present a surgical instrument-state logger that clips onto a conventional instrument without modifying the part that enters the patient and fuses a six-axis IMU and a time-of-flight (ToF) rangefinder with a markerless camera in an error-state Kalman filter under the remote center of motion (RCM) of the trocar. Heading comes from the shaft silhouette, segmented by a U-Net, in place of the magnetometer: the rotation-angle error is 0.200{\deg}, against 3.58{\deg} from the accelerometer and magnetometer alone. Against a Franka Research 3 manipulator, and without alignment to it, the displacement error over 300 translation trials was 1.21mm RMS and the relative-rotation error over 180 rotation trials 0.34{\deg} RMS. On continuous trajectories, tracked and displayed in real time, the absolute tip error was 1.22mm (programmed) and 3.04mm (teleoperated) after post-hoc tuning of three filter parameters, and the full fusion beat every sensor subset. Because the estimator uses no magnetic measurement, its accuracy does not rely on an undisturbed field. The same clip-on device could thus record metric tip trajectories during routine hand-held laparoscopy, while displaying the insertion depth and attitude that are hidden once the instrument is inside the patient.

We study the estimation of a $K$-dimensional simplex from $N$ i.i.d.\ points sampled uniformly from its interior; the observations are convex combinations of $K+1$ unknown prototypes. Existing polynomial-time estimators need cubic per-sample work or $O(NK)$ storage and are impractical at $N\sim 10^6$–$10^8$. We propose DeepMVSA, which re-expresses the minimum-volume principle in neural implicit form: a lightweight coordinate network generates the mixing weights and a triangular LU-type parameterization the dual simplex matrix, reducing the trainable-state memory to $O(K^2)$, independent of $N$, and the cost per data pass to $O(NK^2)$. We prove a non-asymptotic sample-complexity bound of the polynomial-time benchmark order for a localized surrogate estimator; an oracle inequality for every global minimizer of the neural objective, with volume-inflation control and an explicit shrinkage bias; a conditional end-to-end error budget separating statistical, approximation, optimization, and enclosure-residual terms on an explicit envelope event; and two-point lower bounds: at any noise level $\sigma>0$ fixed independently of $N$, the $N^{-1/2}$ scaling is unimprovable in its $N$-exponent. Experiments with up to $N=10^8$ synthetic observations are consistent with the predicted accuracy and scaling, and feasibility on real scenes of $\sim 10^7$ pixels is demonstrated.

Fine-tuned Large Language Models (LLMs) significantly advance Automated Theorem Proving (ATP), but are often deployed as guiding policies within tree search rather than for single-attempt generation. Recent work shows cross entropy is suboptimal for an LLM used in flat search strategies such as aggregation or filtering and that work has developed new loss functions to correct this misalignment. Extending this alignment to tree search is more challenging: proof discovery depends on exploration and recovery through off-trace states that supervised demonstrations do not reveal. We extend Compute-Aligned Training (CAT) to this setting through an abstraction of policy-guided search, deriving tractable, trace-supported losses. Alongside these search-aware losses, we introduce a search-agnostic uniform-allocation (UA) loss that accounts for the budget without specifying the specific search. Both induce scalar weights on per-tactic cross-entropy gradients. We characterize how off-trace behavior affects the search-aware weights, including conditions for vanishing approximation error at large budgets. On a Lean benchmark, both approaches achieve higher observed proof-success rates than cross-entropy across six search strategies, with strong results from a single shared UA adapter. Budget sweeps show larger gains over cross-entropy at 16 than at 256 expansions, implying CAT scales with test time compute.

We prove the following: If $X$ is a plane separating continuum, then every embedding of $X$ into $\mathbb{R}^2$ contains the vertices of a Euclidean rectangle. We arrive to this result by extending a known result by H. Vaughan for Jordan curves to a wider class of topological objects via shape theory and Steenrod homology.

A quaternary Legendre pair of length $\ell$ is a pair of sequences over $\{\pm 1, \pm \mathrm{i}\}$ whose periodic autocorrelations sum to $-2$ at every nonzero shift. We give a quaternary Legendre pair of length $64$. To the best of our knowledge, this is the first reported construction at this length.

LLM-based user simulators aim to bridge the offline-online gap in recommender evaluation by emulating users through injected traits, where preference attributes determine what a user engages with and a behavioral activity trait governs how long they browse. However, we show this intended trait independence collapses during simulation, causing two failures: (i) Trait Interference, where amplified activity distorts preference boundaries and forces interactions with mismatched items to sustain browsing, and (ii) Evaluation Invalidity, where satisfaction scores inflate with activity-driven page counts despite taste mismatches, biasing evaluation toward trait distributions rather than recommender performance. To resolve this, we propose PQA, a page-level quality anchoring method that guides simulators using a personalized anchor reflecting each user's intrinsic preference standard. By assessing whether a page meets this standard before further browsing, PQA enables proactive exits from low-quality pages, letting the activity trait retain its intended role of modulating browsing depth within preference-conforming pages. Experiments show PQA mitigates trait interference and improves the reliability of LLM-based simulator evaluation under activity shifts. Our code is available at https://github.com/chaehyun1/PQA

The realization of a scalable quantum network (QN) hinges on the efficient distribution of entanglement across heterogeneous hardware platforms. While traditional QN frameworks are often restricted to single-encoding schemes, relying exclusively on either discrete (DV) or continuous (CV) variables, such segregated architectures limit the capacity and interoperability of large-scale systems. Here, we propose a scalable hybrid entanglement distribution scheme that bridges these domains by mapping hybrid entanglement swapping and concentration at DV–CV interfaces onto a unified set of series and parallel graph rules. We demonstrate that our hybrid approach exhibits a {potential} performance advantage over traditional point-to-point generation of maximally entangled states, achieving enhanced end-to-end entanglement within series-parallel topologies. These results provide a theoretical foundation for designing resource-efficient, interpretable architectures for the future quantum internet.

Large language models (LLMs) exhibit a parametric vulnerability to adversarial swarm consensus. To mitigate this sycophancy, we introduce Contrastive Epistemic Decoding (CED), a zero-shot inference intervention. Unlike standard Contrastive Decoding (CD) which relies on a weaker secondary model, CED utilizes a dual forward-pass on a single architecture to isolate conformity bias. By introducing a novel asymmetric, zero-bounded probability clamp and discrete top-k truncation mask, CED mathematically suppresses toxic consensus tokens without causing grammatical collapse. Evaluated across 7,200 paired trajectories on complex benchmarks (GAIA, SWE-bench, Multi-Challenge) using Gemma-2 (9B), Llama-3.1 (8B), and Mistral v0.3 (7B), CED successfully neutralizes architectural and positional biases. By reducing cognitive loafing by up to 33.00% absolute, CED drives significant performance gains, yielding up to a 30.75% accuracy recovery. Regaining sovereignty induces distinct architectural behaviors—passive task-focus in Gemma-2 and active refutation of the simulated swarm in Llama-3.1—showing CED decouples compliance from capability without fine-tuning.

Soccer tactics are interactive: an attacking action changes the opponent's defensive problem, and the observed response depends on the multi-agent match state. We introduce SambaGraph, an action–reaction spatio-temporal graph dataset and benchmark for soccer tactical response modeling. From tracking and event data for all 64 matches of the 2022 FIFA World Cup, we curate 4,070 action-centered episodes represented as temporally aligned 23-node player–ball graph sequences with attack/defense views, response labels, and 26,270 split-safe attack–defense pairs. We study three questions: whether observed responses can be classified from graph episodes, whether successful defenses can be retrieved for a query attack, and whether graph-derived summaries support grounded LLM reasoning. A compact signature MLP obtains $0.796\pm0.007$ macro-F1 for response classification, while a fused graph–signature dual encoder reaches $0.471\pm0.029$ Hit@5 and $0.655\pm0.051$ Hit@10 for full-bank defensive retrieval. Hard negatives maximize pair discrimination but not retrieval quality. Local LLMs underperform supervised encoders for direct classification and do not improve over a strong original order in eight-candidate reranking, but they provide grounded tactical rationales. These results position SambaGraph as a reproducible benchmark for graph-based soccer strategy-response research. Code and dataset are available at: https://github.com/areyesan/SambaGraph.

Let $0<\kappa<1$ and set $p_+=2/(1-\kappa)$ and $p_-=2/(1+\kappa)$. We construct a coefficient $\kappa\leq a\leq\kappa^{-1}$, smooth outside a compact set of Lebesgue measure zero, for which the $W^{1,2}_{p_+}$ estimate fails for the one-space dimensional nondivergence form parabolic equation. The corresponding solution has second spatial derivative in the weak $L_{p_+}$ space, but not in $L_{p_+}$. By duality, the $W^{1,2}_{p_-}$ a priori estimate also fails. Conversely, we demonstrate that the $W^{1,2}_p$ estimate holds and the equation is uniquely solvable for $|p-2|<c\kappa$, showing that the size of the solvability interval around $2$ has optimal order $\kappa$ as $\kappa\downarrow0$, which addresses a question raised in [17]. These results imply that the Alexandrov maximum principle holds for $p>2-c\kappa$ and this order is sharp as $\kappa\to 0$. The corresponding results for divergence form equations are also obtained.

A lack of high-throughput phenotyping solutions for root traits in field-grown crops has severely constrained understanding and improvement of below-ground traits and processes. Minirhizotrons are the standard non-destructive root-phenotyping method in field environments. Computer vision solutions are needed to allow automated trait estimation at scale, but training data is scarce and human annotations are often inaccessible because they reside in proprietary software that only exports per-image scalar totals of root length and surface area. Nevertheless, large numeric archives of these root traits already exist. RootQuant showed that the traits can be predicted directly from the whole image by regression, thus removing manually traced masks from the pipeline; RootQuantV2 takes that idea further by replacing RootQuant's CNN backbone with a self-supervised ViT. We adapt a frozen DINOv3 ViT-L/16 with a hybrid parameter-efficient scheme. Training only 11.9M parameters (3.78% of the model), RootQuantV2 achieves length and area $R^2$ of 0.950 and 0.930, respectively, while lowering length/area RMSE by 24.3%/20.7% over RootQuant. RootQuantV2 thus repurposes legacy numeric archives for high-throughput, automated root trait estimation.

Time-synchronized state estimation (SE) plays a critical role in ensuring real-time situational awareness in modern power systems. However, achieving full system observability using phasor measurement units (PMUs) is often impractical due to cost and deployment constraints. Moreover, SE operation at PMU timescales imposes stringent requirements on latency, robustness, and reliability that are difficult to satisfy using conventional iterative hybrid SE techniques under incomplete observability by PMUs. This paper evaluates the feasibility of deploying deep neural networks for PMU-timescale, time-synchronized SE in real-world PMU-unobservable transmission systems using actual data from a US power utility. Key contributions include a systematic assessment of estimation accuracy, scalability, and computational performance under realistic operating conditions.

We investigate axial gravitoelectromagnetic perturbations of charged black holes surrounded by dust-like Kiselev-type anisotropic matter in Rastall gravity. We derive the axial matter compatibility condition and construct a compatible response by holding the covariant anisotropy-direction vector unperturbed. The resulting gravitational and electromagnetic perturbations obey two coupled Schrodinger-type master equations. An r-independent algebraic decoupling exists in the Reissner-Nordstrom and general-relativistic dust-like limits; additional radial structure obstructs it in generic charged Rastall backgrounds with nonzero surrounding matter. We prove axial mode stability under the adopted closure for every nonextremal background satisfying the stated exterior conditions, covering all physical axial multipoles. A matrix S-deformation excludes exponentially growing coupled modes for l >= 2, while the physical electromagnetic dipole has a positive effective potential. A Chebyshev pseudospectral calculation, validated against the Reissner-Nordstrom spectrum and spectral convergence, determines the fundamental frequencies. In the parameter ranges examined, increasing the Rastall coupling lowers both the oscillation frequency and the damping rate in the small-coupling region; in the large-coupling region, the oscillation frequency rises while the damping rate varies nonmonotonically. A larger charge generally raises the oscillation frequency, whereas stronger surrounding-matter contributions produce longer-lived modes.

We construct a sequence of finite Rodriguez-Villegas transforms converging locally uniformly to the Riemann $\xi$-function in the critical strip. The input is a positive symmetric profile on the unit interval obtained from the Riemann theta kernel through convolution with the hyperbolic-secant kernel and the logistic coordinate. The profile is a Stieltjes function of $x(1-x)$. Its Bernstein polynomials produce reciprocal numerators and exact finite functional equations. The same numerators admit an exact realization as fermionic supertraces, while the Bernstein polynomials are normalized Gibbs traces.

Agent memory enables enterprise agents to retain knowledge acquired during work and reuse it across tasks and agents, turning execution experience into persistent organizational knowledge. Realizing this potential requires both source–memory integration, through which enterprise sources and accumulated memory can be utilized together, and memory governance, through which shared memory remains subject to organizational policies throughout its lifecycle. These requirements interact when information from enterprise sources persists in memory. As this information is repeatedly derived and reused under changing principals and policies, source restrictions may be bypassed, resulting in information leakage. Preventing such leakage requires authorization continuity, under which source restrictions remain effective throughout source-to-memory and memory-to-memory derivation and reuse. Existing approaches address these concerns individually, but do not treat source–memory integration, memory governance, and authorization continuity as combined core design targets across the memory lifecycle. We define Governed Enterprise Memory as agent memory designed around this combined scope and present AkasicMEM as its realization. AkasicMEM realizes authorization continuity through transitive lineage, policy composition during memory formation, and policy re-evaluation during retrieval. It is built on GraphAI's AkasicDB, a unified vector–graph–relational database whose storage and execution substrate enables the underlying operations of these mechanisms to be jointly optimized and executed.

Open robot policies increasingly follow two paradigms: vision-language-action models (VLAs) directly map observations and instructions to actions, whereas world-action models (WAMs) incorporate learned video or world dynamics into policy learning or action generation. Although both target the same manipulation tasks and represent alternative design choices, they are commonly reported under different evaluation protocols, leaving their capability, robustness, language sensitivity, and deployment-cost trade-offs unclear. We present IndustrialVLA-Bench, an evidence-aware evaluation of six released VLA and WAM systems under a unified reporting schema. It separately evaluates clean capability on LIBERO, non-language robustness on LIBERO-Plus, instruction sensitivity on LIBERO-Para, and observed execution cost. Reported task scores aggregate three complete evaluations with distinct random seeds under a fixed checkpoint and inference configuration. Across all six systems, clean LIBERO averages differ by only 1.58 points, whereas robustness and paraphrase summaries span 14.62 and 31.08 points. Restricting every comparison to the three protocol-faithful systems preserves the effect (1.36, 14.62 and 23.10 points), so the diagnostic separation reported here does not depend on the weaker evidence tiers. We additionally report observed inference latency, peak memory, runtime mode, and an evidence status for every system. Protocol-faithful, near-reproduction, and pending-verification entries remain visibly separated; only protocol-faithful entries support strict comparisons. Rather than claiming universal superiority of either paradigm, IndustrialVLA-Bench provides traceable evidence for comparing released robot policies on shared practical criteria. Code and evaluation records are available at https://github.com/xiaoqi-7/IndustrialVLA-Bench.

The safe operation of a humanoid robot critically relies on accurate estimation of its vertical orientation, or “tilt”. This requires a variety of mathematical tools, including three-dimensional geometry and differential equations. To formally verify tilt estimation, we develop a library for stability analysis in the Rocq prover. We start by formalizing a theory of ordinary differential equations. We provide a formalization of the (local) Cauchy-Lipschitz (a.k.a. Picard-Lindelof) theorem for existence and uniqueness, taking advantage of the library support for quotients provided by the Mathematical Components library. We extend this formalization with a variant for global existence and continuous dependence on initial conditions. Building on these foundations, we develop a theory of Lyapunov stability that is compatible with an existing formalization of LaSalle's invariance principle. We also extend an existing library for the statics of robot manipulators to support dynamical systems. Finally, we apply these libraries to the verification of a state-of-the-art tilt estimation developed for a humanoid robot.

Large-model parameters have grown beyond the capacity of a single xPU, dispersing across multiple xPUs spanning distinct host domains, where xPU-to-xPU communication dominates overall system efficiency. Existing scale-up interconnect remains inadequate: network-based solutions built on Ethernet–such as RoCE (RDMA over Converged Ethernet)–introduce specific message-semantics and protocol-stack characteristics, and rely on fragmented per-host addressing, while conventional host-based fabrics are confined to a single host domain and lack cross-host unified addressing. This paper presents Co-Fabric, a bus-based interconnect that, unlike conventional bus designs, breaks host-domain boundaries to deliver unified xPU interconnection for scale-up superpods. Co-Fabric makes three contributions: a streamlined four-layer protocol stack achieving nanosecond-scale processing latency with native reliability; a cross-domain scaling and P2P mechanism that routes using port identifiers embedded in the packet header; and a unified address space built on shadow-device auto-enumeration. On a 64-xPU 3D-Mesh system, Co-Fabric cuts inter-node communication latency by over 50% and improves bandwidth by 2-5x over RoCE, accelerating DeepSeek R1 inference by 30%-80%. Moreover, since its streamlined four-layer protocol stack and higher data-communication efficiency reduce protocol and processing overhead relative to the Ethernet-based RoCE stack, Co-Fabric cuts the cost and power of the interconnect itself by up to 80% and 5%, respectively. These results demonstrate Co-Fabric's advantage for AI computing centers.

We investigate and characterize order automorphisms on the set of partial isometries in the finite-dimensional matrix algebra $M_n(\mathbb{C})$. Different from the classical order automorphisms of subspace lattices, which can be implemented by standard invertible or unitary transformations, the order automorphisms considered herein admit no such conventional matrix representations. Instead, they are essentially governed by matrices such that $I-(A+A^*)$ is either positive or negative invertible. The results reveal that the structural features of order automorphisms for partial isometries are substantially more intricate than those of classical subspace automorphisms.

Vision-language-action policies benefit from geometric supervision, but current-frame geometry alone does not explicitly describe the changes associated with manipulation. This design is motivated by the goal of learning an embodiment-agnostic visual interface that can be pretrained across robot and egocentric video before robot-specific action alignment. We introduce Geometry-Change VLA (GC-VLA), which learns to predict multiview future-current geometry-change tokens from current observations. Offline frame pairs define a nominal 0.5-second prediction horizon; future observations are used only to construct training targets. Stage 1 trains a geometry-change vision-language model (GC-VLM). Stage 2 introduces a continuous ActionExpert and aligns it with robot actions while stopping action-flow gradients at the VLM interface. Stage 3 enables these gradients to update the trainable VLM components jointly with the ActionExpert. Stage 4 freezes GC-VLA and applies Geometry-Conditioned Residual Flow (GCRF), using a binary intervention router and a single bounded residual velocity policy learned from closed-loop feedback. GC-VLA achieves 95.20% success on LIBERO, and GC-VLA with GCRF achieves 99.55%. Inference uses current observations and the learned GC representation without executing the offline target encoders.

We define a signed GLMY chain complex over $\mathbb{R}$ for signed digraphs using sheet-labelled regular paths. The complex is naturally isomorphic to the deck anti-invariant subcomplex of the ordinary GLMY complex on the signed double cover. The double-cover realization yields switching invariance and recovers ordinary GLMY homology for switching-balanced signings. Bidirected completion gives an orientation-independent homology theory for signed graphs. For a signed graph, the zero-dimensional homology identifies with the kernel of the signed Laplacian and has dimension equal to the number of balanced connected components. Signed GLMY homology is functorial under signed weak morphisms, which combine vertex maps with switching functions and allow compatible arrow contractions. For signed digraphs, the all-positive reduction retains the orientation sensitivity of ordinary GLMY homology, while explicit computations show additional sensitivity to the arrow signs. For a fixed digraph with five vertices and nine arrows, we classify all 512 arrow signings and obtain exactly four signed Betti vectors. Precisely 16 signings have nonzero second signed GLMY homology.

In their studies of pathologies in recursion categories, Montagna (1989) and Di Paola–Montagna (1991) introduce the algebraic systems $S'$ and $S'_T$, respectively, and claim that they are categories. We show that the proposed composition is not independent of the choice of representatives. For every consistent recursively enumerable extension $T$ of Peano arithmetic ($\mathrm{PA}$), we exhibit two unary programs whose partial functions are provably equal in $T$, separately at each standard input. Composing each after a program that searches for a $T$-proof of contradiction and returns its code yields programs that are not equivalent in this sense. An alternative proof uses the productivity of the complement of the diagonal halting set. Montagna's $S'$ is the case $T=\mathrm{PA}$. More generally, for consistent $T\supseteq\mathrm{PA}$, pointwise provable equality is a composition congruence exactly when $T$ proves every true $\Pi^0_1$ sentence, in which case it is extensional equality. This completeness condition fails for every consistent recursively enumerable $T\supseteq\mathrm{PA}$ by Gödel's second incompleteness theorem. For every extension $T\supseteq\mathrm{PA}$, the least composition congruence containing pointwise provable equality is extensional equality if $T$ is $\Sigma^0_1$-sound and the universal relation otherwise.

Supervised approaches to quantum error mitigation learn a map from noisy circuit outputs to ideal ones, and therefore require the ideal outputs. Producing those ideal outputs demands noiseless classical simulation, whose cost grows exponentially with system size, so supervision is unavailable in exactly the regime where mitigation matters most. We ask whether cheap, individually unreliable signals drawn from circuit structure and hardware calibration can take the place of ideal labels. We assemble sixteen heuristic labeling functions (stabilizer and parity constraints, relaxation and readout characteristics, local depth, gate counts, and neighboring activity), reconcile their disagreements with a probabilistic label model, and read the resulting per-qubit error probabilities as a readout channel whose inverse mitigates the measured distribution. No ideal output enters the training path. On $147{,}000$ five-qubit circuits executed on two IBM devices, the method removes $24.3\%$ (Algiers) and $28.8\%$ (Hanoi) of the Kullback-Leibler divergence to the ideal distribution, against $15.4\%$ and $21.5\%$ for the strongest published analytical baseline, a margin that holds on both devices and lies far outside its bootstrap interval. Supervised neural models trained on ideal distributions remain stronger where such labels exist, and we quantify that gap rather than setting it aside; the method's claim is to the regime where they do not, since the labels they require cannot be computed for the circuits mitigation is needed for. The codes will be released shortly.

We propose a two-stage production method to overcome existing supply-constraints for the alpha-emitter $^{149\mathrm{g}}$Tb, a promising candidate for Targeted Alpha Therapy (TAT) with no existing globally scalable production pathway. Although awaiting experimental measurement of the $^{150}$Gd(p,2n)$^{149\mathrm{g}}$Tb cross section, the proposed method could produce $^{149\mathrm{g}}$Tb at clinical scale and beyond on readily available proton cyclotrons, enabled by production of the extinct but long-lived isotope $^{150}$Gd, a pure alpha emitter with a 1.79 million year half-life. Stage one generates $^{150}$Gd feedstock by irradiating natural Eu or enriched $^{151}$Eu with $\gtrsim$10 MeV protons, neutrons or photons. Stage two produces $^{149\mathrm{g}}$Tb from fabricated $^{150}$Gd targets by driving the $^{150}$Gd(p,2n)$^{149\mathrm{g}}$Tb reaction with $\gtrsim$14 MeV protons, accessible on over 700 reported cyclotrons worldwide. Fast fusion neutrons appear to offer the most scalable pathway for $^{150}$Gd production: even with a large $^{149 \mathrm{g}}$Tb dose size of 1 GBq and 40 million administered doses/yr, we estimate this would require neutrons produced by only 6.8 megawatts of steady-state deuterium-tritium power to produce the required $^{150}$Gd, far below expected capacity in the next decade. The route described here, if validated, would enable $^{149\mathrm{g}}$Tb supply at the scale needed to support clinical development of $^{149\mathrm{g}}$Tb-based TAT.

We study the supercritical LYZ equation on compact Kähler manifolds at the boundary of stability. Under numerical semistability, we prove a quantitative positivity estimate on the modified nef cone. It implies that the destabilizing prime divisors form a finite exceptional family. Assuming the existence of a smooth semisubsolution, we prove that the associated intersection form is negative definite on the span of classes of these divisors. Using this rigidity, we then construct a logarithmic singular subsolution along their union and obtain a bounded Bedford–Taylor solution which is smooth on its complement.

In connection with eigenvalue perturbation bounds, Li [Linear Algebra Appl., 278 (1998), pp. 317-326] formulated three minimization problems involving Hadamard products. The third asks how to relate $$ \min_{W\ \mathrm{nonsingular}}\|W\circ G\|_\mathrm{ui}\,\|W^{-1}\circ G^\mathrm{T}\|_\mathrm{ui} \quad\mbox{and}\quad \min_{P\ \mathrm{permutation}}\|P\circ G\|_\mathrm{ui}^2, $$ where $\|\cdot\|_\mathrm{ui}$ denotes a unitarily invariant norm and $\circ$ denotes the Hadamard product, i.e., entrywise multiplication. The first quantity never exceeds the second, and a reverse inequality with a constant independent of the matrix size was conjectured. Romeo and Tilli [Linear Algebra Appl., 326 (2001), pp. 161-172] showed that if such a bound holds for every $G$, then $\|\cdot\|_\mathrm{ui}$ must be uniformly equivalent to the Frobenius norm $\|\cdot\|_\mathrm{F}$. We prove that the two quantities coincide when $\|\cdot\|_\mathrm{ui}=\|\cdot\|_\mathrm{F}$: for every complex matrix $G$, $$ \min_{W\ \mathrm{nonsingular}} \|W\circ G\|_\mathrm{F}\,\|W^{-1}\circ G^\mathrm{T}\|_\mathrm{F} =\min_{P\ \mathrm{permutation}}\|P\circ G\|_\mathrm{F}^2. $$ The associated conjecture therefore holds with the optimal constant $c=1$.

We study Keller–Segel systems on $ \mathbb{T}\times\mathbb{R}$ subject to a stochastic Couette transport flow. We prove that sufficiently strong mixing suppresses chemotactic finite-time blow-up with high probability; thus yielding a unique global-in-time mild solution for initial data of arbitrary mass. Moreover, we obtain a quantitative estimate for the probability of blow-up, showing that it decays exponentially with increasing strength of the stochastic shear. Our analysis is based on the enhanced dissipation generated by the stochastic shear flow. To prove our main result, we first construct a maximal local mild solution through pathwise estimates for the random solution operator of passive scalar transport equations and a fixed-point argument. To obtain global control, we decompose the evolution into time blocks and exploit mixing-induced decay of the nonzero spatial Fourier modes. A nonlinear bootstrap argument then yields global existence above an almost-sure finite random threshold for the shear strength which is obtained by an application of the Borel-Cantelli theorem.

We prove a Positive Mass Theorem for $C^0$-asymptotically flat Riemannian metrics with nonnegative scalar curvature in a weak sense that are sufficiently uniformly close to Euclidean space. More precisely, we show that a $C^0$-asymptotically flat Riemannian metric that is a $C^0$ perturbation of Euclidean space with nonnegative scalar curvature in the sense of Ricci flow has nonnegative mass, where the mass is given by a $C^0$ analog of the classical ADM mass previously introduced by the author.