Papers

Alman and Vassilevska Williams (arXiv:2610.06783) gave the first polynomial improvements over the textbook algorithms for 3SUM and APSP: deterministic algorithms with exponents 1.9992 for 3SUM, 2.9983 for Exact Triangle and 2.99942 for APSP. Their algorithm has two steps: a matrix theorem, computing N^2/sqrt(D) prescribed entries of the product of an N x D and a D x N integer matrix in time N^2/D^gamma, gamma > 0, when D <= N^eps, for every eps < 0.1204; and reductions from the three problems to the lopsided sparse triangle problem. We improve both. With Schönhage's five-term partial-product tensor, the matrix theorem holds for every eps < 0.172. The preprint's reductions keep a quarter of the matrix theorem's saving for 3SUM and a half for Exact Triangle and APSP; ours keep all of it, by calling the counting version of the triangle problem adaptively and building composite moduli one prime at a time. Together this gives deterministic 3SUM in n^{1.9945+o(1)}, and Exact Triangle, (min,+)-product and APSP in n^{2.9945+o(1)}. We also improve on the preprint's proof sketch for 3XOR and solve it deterministically in O(n^{1.99451}). Which other problems on three lists does the method make truly subquadratic? It solves 3SUM over F_p^d for every fixed prime p with 3XOR's exponents, and over every finite abelian group of exponent p^2, such as Z_4^d, in O(n^{1.99726}). For coordinatewise problems with a fixed relation P on every coordinate triple, a certified exhaustive computation shows that every Boolean P, and every functional P over three letters, either is the solution set of an equation over Z, F_2 or F_3, and the problem is then truly subquadratic, or has a gadget for Orthogonal Vectors, and the problem is then hard under the Orthogonal Vectors hypothesis. Over four letters, just four functional relations, up to symmetry, satisfy neither condition.

Autonomous quadrotor flight in cluttered environment is an key capability for tasks like inspection and search. Navigation policies trained in simulation and transferred to hardware can fly at high speed in open space, but they can still collide as obstacle clearance shrinks. We attribute most of these collisions to a single mechanism, namely motion in directions the onboard camera does not observe. Heading that is not aligned with the direction of travel and lateral excursion during obstacle avoidance both carry the vehicle toward obstacles outside its field of view. We introduce ClearNav, a navigation policy trained with two reward terms that constrain motion to the observed direction, a forward-looking heading target and a penalty on lateral velocity. A third term penalizes speed near obstacles so that an obstacle entering the field of view remains avoidable. In simulation, across 18 held-out scenarios with 2304 paired flights per policy unseen in training, ClearNav improves success by 6.8 points, cuts the collision rate by 65% relative to the state of the arts, and flies at maximum 5m/s. For the proximity term alone under matched commanded speeds, the collision reduction still holds at 70%. The effect is significant on static scenes and consistent in direction on moving obstacles, though not significant there at the suite level, showing that safer flight in clutter is possible even on the limited sensors and compute of small drones. We also validate our navigation policy in a real-world cluttered environment, showing that it enables the quadrotor to navigate through the cluttered environment successfully and safely.

Agentic workflows combine large language models with planning, memory, tools, execution environments, and iterative feedback. Their failures are difficult to diagnose because the same user-visible symptom may originate from model behavior, workflow orchestration, external tools, infrastructure, or their interaction. This paper presents a source-aware taxonomy of agentic workflow failures and applies it to evidence from academic literature and GitHub. The taxonomy combines source attribution with three empirical dimensions: a concrete failure-type dictionary, workflow-stage localization, and agent-capability localization. Its central structure is a two-axis localization matrix: workflow stages form the horizontal process axis and agent capabilities form the vertical functional axis. This design moves beyond taxonomies that mainly classify final failure manifestations by localizing intermediate process states inside agent operation. Our empirical comparison shows that literature-derived failures emphasize planning, reasoning, learning, perception, and knowledge-related mechanisms, whereas GitHub-derived failures concentrate on execution-stage, actuation-related, tool, API, and infrastructure failures. The dominant hotspot is iterative execution and actuation. These findings indicate that agentic workflow reliability should be treated as a systems problem rather than only a model-output problem. The taxonomy provides a basis for structured evaluation, root cause analysis, failure injection, and engineering triage.

Understanding the functionalities of heterogeneous catalysts requires high-resolution nanoscale chemical imaging tools. X-ray spectro-ptychography offers this capability by enabling X-ray absorption near edge structure (XANES) based amplitude and phase contrasts imaging with high spatial resolution. However, such demonstrations at energies above 10 keV, such as the Au L-edges, remain limited due to weak X-ray absorption cross-sections and correspondingly, low signal levels. Here, multimodal spectro-microscopy has been performed on Au microcrystallites that host both conventional face-centered cubic (fcc) and unconventional non-cubic lattices, particularly important for catalysis. Amplitude and phase contrasts across the $\mathrm{L}_3$ edge are resolved on thin Au crystallites (150 nm to 1.71 $\mu\mathrm{m}$) with 45 nm spatial resolution. The resulting XANES spectra exhibit excellent agreement with standard transmission and fluorescence contrasts, while providing complementary chemical insights with improved spatial resolution. Quantitative retrieval of anomalous phase variations ~12 keV further demonstrates the suitability of this method for high-resolution chemical imaging across a wide range of functional materials.

I study a dynamic pricing problem in which a seller with limited inventory must sequentially price a perishable product for arriving customers with heterogeneous valuations. The key difficulty lies in balancing immediate sales against future opportunities under capacity constraints and stochastic customer valuations. I propose a simple, interpretable, and asymptotically optimal dynamic pricing algorithm, called Dynamic Run-Out Pricing (DROP), that adjusts prices in real time based on current inventory and anticipated future demand. Under the assumption that customers' valuations follow bounded or exponential distributions, our theoretical results show that DROP's expected revenue converges to the optimal revenue as market size grows. Unlike traditional notions of near-optimality, where the ratio of the algorithm's revenue to the optimal revenue converges to 1, I show that the absolute difference between the two revenues converges to 0 as the market size increases, underscoring the algorithm's effectiveness. Furthermore, I extend our analysis to heavy-tailed valuation distributions, specifically the Pareto distribution, where standard run-out heuristics severely underprice inventory by ignoring the value of waiting for extreme valuations. To correct this, I introduce the Tail-Aware DROP (TA-DROP) heuristic, which intentionally inflates prices to protect inventory, and I prove that it restores asymptotic optimality with a strict polynomial convergence rate.

The Leavitt & Pickering (1912) mean period-magnitude relation for SMC Cepheids exhibits correlated residuals relative to modern Gaia $B_G$ and Johnson $V$ photometry ($r_{\delta V/\delta L12}\approx0.76$). Therefore dispersion inherent to Leavitt's photographic photometry relation is not solely random scatter ($\sigma_{L12,B_G,V}=0.22\pm0.03,0.25\pm0.04,0.20\pm0^{m}.03$), and is potentially consistent with the partial encoding of combined effects from the elongated SMC, classical Cepheid instability strip, differential reddening, etc. OGLE data indicate that $\gtrsim870$ SMC variables identified by Leavitt (1908, Table 2) are classical Cepheids ($\gtrsim 90$ %), and the preliminary unphased Table 2 photometry likewise features correlated residuals relative to contemporary observations ($r_{\delta B_G/\delta L08-T2}\approx0.54$). Table 6 data in Leavitt (1908, $n=16$) define a highly linear Leavitt law over an extensive baseline ($r\approx-0.98$), but constitute a smaller sample than the later 1912 work ($n=25$).

The distribution of energetic neutral atom (ENA) differential flux density at the heliospheric boundary is the key to revealing the interaction mechanism between the solar wind and the interstellar medium. The Helios ENA Rapid Mapper (HERMA) is a pre-research project for a new ENA detection concept and technical principle, which is designed to provide technical support for refined detection with its advantages of wide energy range coverage and high spatial resolution. To verify the high spatiotemporal resolution rapid imaging capability of the HERMA, this study restored the particle response characteristics of the HERMA (spatial modulation grid + 256-pixel linear array silicon detector) based on Geant4 simulation, and we compared the performance of the traditional inversion algorithm with the convolutional neural network (CNN) inversion algorithm using INCA data as the benchmark. The results show that under different sampling times, the source distribution inverted by the CNN algorithm has a higher degree of agreement with the ideally expected distribution, and the CNN algorithm exhibits better stability and accuracy than the traditional algorithm under the conditions of low signal-to-noise ratio and short staring time. This study provides core technical support for the engineering verification and on-orbit application of the HERMA instrument, fills the technical gap in wide-energy-range and high-resolution ENA detection, and helps to deepen the research on the interaction mechanism between the solar wind and the interstellar medium at the heliospheric boundary.

Cylindrical or spherical windows often appear in experiments and real applications. Measuring through those windows requires projecting 3D points to pixels, which is different to compute due to refraction. A point may have multiple projected images but cannot be efficiently determine with the existing projection algorithms. Using the ray's conserved momenta, we reduce this projection to closed-form equations with a concave structure. As a result, we can successfully determine whether a 3D point has a single projection or multiple and compute those locations. In simulations, with one single search, we can recover images that iterative methods miss even with many restarts. A wand calibration based on it can reach the precision limit set by image noise, enabling accurate 3D particle tracking in a swirling flow through cylinder windows. This algorithm shows how to predict visibility and multiple images through curved windows, shedding light for 3D pipe flow measurements, accurate underwater reconstruction, and many real applications.

Vision Transformers incur high computational costs when processing long visual token sequences. Existing reduction methods typically rank tokens independently and discard those deemed unimportant or redundant. We propose a reconstruction-guided framework for visual token condensation (ReConV), which summarizes the complete visual sequence with a compact Summary Bank (SMB) and retains tokens whose information is not adequately captured by the SMB, as indicated by their reconstruction residuals. At each compression layer, a lightweight position-conditioned decoder reconstructs the current tokens, leaving shared information compressed in the SMB while explicitly preserving poorly reconstructed details. Experiments on image classification and multimodal understanding demonstrate favorable performance-efficiency trade-offs: ReConV surpasses the uncompressed baseline with 48% fewer GMACs on ImageNet and enables reusable image-level representations for efficient multi-turn inference. Code is available at TokenDrop.

Using the recent single-generation theorem for von Neumann $\mathrm{II}_1$ factors with separable predual, we prove the following results on the generators of a separable unital C*-algebra. Let $A$ be a separable unital C*-algebra, and suppose that $\partial \mathrm{T}(A)$ is nonempty, compact, and zero-dimensional. Then (1) If $A$ has strict comparison of positive elements by tracial states, then $A$ can be generated by two ordinary elements. (2) If $A$ has real rank zero and $A$ has strict comparison of projections by tracial states, then $A$ is singly generated. In this case, its single generators form a norm-dense $G_\delta$ subset of $A$. (3) If $A$ has strict comparison of positive elements by tracial states and there is a unitary $u\in A$ such that for every $\tau\in \partial \mathrm{T}(A)$, (a): the spectral distribution of $u$ with respect to $\tau$ is atomless and has full support on $\mathbb T$, and (b): for every $a\in A$, $$\lim_{t \to 0}\sup_{p\in D_\tau, 0<\tau(p)\leq t} \| p\bigl(\pi_\tau(a)-\mathbb E_{D_\tau}(\pi_\tau(a))\bigr)p\|=0, $$ where $M_\tau=\pi_\tau(A)''$, $D_\tau=\pi_\tau(u)''$, $\mathbb E_{D_\tau}: M_\tau \to D_\tau$ is the trace-preserving conditional expectation, and $\pi_\tau$ is the GNS representation associated to $\tau$, then $A$ is singly generated. In particular, Conclusion (3) above implies that the reduced group C*-algebra $\mathrm{C}_r^*(\mathbb F_n)$ is singly generated for every $n\in\{2,3,\ldots,\infty\}$.

Advances in large-scale machine learning have enabled agentic AI systems: persistent, autonomous computational entities that pursue goals, initiate actions, and interact with open environments. These systems are often framed as applications of foundation models. We argue that this framing is incomplete: agency is also a candidate new organizing abstraction for computer science. What is new is not the concept of an agent, nor runtime policy generation per se, but its general-purpose, open-ended form, in which systems can synthesize courses of action from goals, observations, and memory, over tools and environments for which no complete task-specific model was specified in advance. Thus agency becomes a systems-level abstraction as well as an AI modeling concept. Where the process abstraction organizes computation around executing programs, agency organizes it around pursuing goals under bounded resources: a process executes a program; an agent pursues a goal. From this perspective, the recurring failures of deployed agentic systems (runaway resource consumption, behavior redirected by adversarial content, opaque continuous operation) are symptoms of structural abstraction gaps, not isolated engineering defects. We examine these gaps across six dimensions (execution, interfaces, correctness, oversight, coordination, and persistence) and argue that closing them requires coordinated advances in systems,

We argue that cooperative multi-agent reinforcement learning (MARL) should treat test-time coordination as part of the problem definition rather than a deployment artifact. Our focus is on long-horizon, centrally monitored cooperative systems such as warehouse robotics, automated storage-and-retrieval systems, and cloud-resource scheduling, where teams share a reward, operate under shifting objectives, and already run with global telemetry at deployment. The dominant paradigm in this regime, centralized training with decentralized execution (CTDE), uses global information only during training and provides no coordination channel at execution. This is sufficient when agents are weakly coupled or the environment is stationary, but it fails when the team must realign after conditions change. The limitation is structural rather than algorithmic, and more training cannot add a channel that does not exist at execution time. We call this the recoverability gap. To address it, we name the underexplored design space of intermittent test-time coordination as centralized training with mixed execution (CTME), in which a coordinator observes global state and emits high-level signals while agents act locally. CTME is not a particular learning algorithm, but a cooperative MARL problem setting that makes test-time information structure explicit. It specifies team-level information, high-level signals, and update frequency after training, with CTDE recovered as a limit case. We do not argue that CTDE is obsolete. We argue that removing the test-time channel is a modeling choice, not a necessity, and the wrong default when global telemetry is available and teams must recover from drift.

The finite-dimensional $F$-algebras $D=F(e_{11}+e_{33})+Fe_{22}+Fe_{12}+Fe_{13}+Fe_{23} \subseteq UT_3$ and $D_0=Fe_{22}+Fe_{33}+Fe_{12}+Fe_{13}+Fe_{14}+Fe_{23}+Fe_{24}+Fe_{34} \subseteq UT_4$ occupy a distinguished position in the theory of central polynomials. Together with the infinite-dimensional Grassmann algebra, these algebras generate the only varieties of almost polynomial growth of proper central codimensions. We determine finite generating sets for their $T$-spaces of central polynomials and compute their central and proper central codimensions. We also prove that every group grading on $D_0$ is isomorphic to an elementary grading and determine the identities and codimensions associated with all its $\mathbb{Z}_2$-gradings.

Generating realistic time-series requires faithfully reproducing rare but high-impact extreme events. Existing approaches often rely on specialized representations or extreme-focused training objectives, which may restrict the diversity of captured extremes or compromise overall distributional fidelity. We propose Cross-View Time-Frequency Diffusion (CrossTF), a framework that exploits discrepancies between time- and frequency-view denoisers for extreme-aware time-series generation. Our key empirical finding is extreme-sensitive cross-view asymmetry: moving along the direction connecting the two views' clean-sample estimates substantially changes extreme fidelity while having limited effects on overall generation quality. We further identify a late-stage cross-view window in which this discrepancy stabilizes and cross-view adjustments have a particularly strong effect on extreme fidelity. These findings motivate a staged optimization strategy in CrossTF: after training and freezing both denoisers, it only needs to learn step-specific fusion parameters within this window using an Extreme Value Theory-based objective under an energy-distance constraint. This formulation supports interpolation between and extrapolation beyond the view-specific estimates while limiting degradation in overall generation quality. Extensive experiments on seven real-world datasets against 11 baselines show that CrossTF improves extreme fidelity by 26.1% and overall performance by 7.4% relative to the strongest baselines. These results highlight cross-view discrepancy as an effective basis for low-dimensional optimization in extreme-aware time-series generation.

On the Internet, we use the Base64 format to represent binary data (e.g., images or cryptographic keys) as text. The ES2026 JavaScript standard, following the WHATWG forgiving-base64 rules, requires decoders to ignore ASCII whitespace while rejecting other non-alphabet characters. Prior SIMD Base64 decoders assume whitespace-free input. We present a block-oriented decoding algorithm that removes whitespace inside the SIMD loop: clean 64-character blocks are decoded directly, while blocks with ignorable characters are compacted into a small stack buffer. We provide implementations for x64 (SSSE3 through AVX-512 VBMI2), ARM NEON, and LoongArch. Our C++ decoder reaches up to 94 GB/s on recent AMD Zen 5 processors. In Node 25, integration of our decoder raises Uint8Array.fromBase64 throughput by up to 6x compared with Node 19. The implementation has been adopted by major browsers and JavaScript runtimes.

A group of conformal automorphisms of a Riemann surface is said to act purely non-free, or to produce a gpnf-action, if each of its finite-order elements has fixed points. It is known that every finite group admits a gpnf-action on a suitable closed Riemann surface of genus at least two. If $G$ is an infinite countable group, and $X$ is an infinite-type orientable surface, without planar ends and whose space of ends is self-similar, then we observe that there is a Riemann surface structure on $X$ admitting a gpnf-action by $G$. We obtain upper bounds for those finite groups producing a gpnf-action on a closed Riemann surface of genus $g \geq 2$. If $G$ is a finite group, then its purely non-free genus $\sigma^{pnf}_{hyp}(G)$ is the minimum genus of at least two of a closed Riemann surface admitting a gpnf-action by $G$. This parameter is already known for cyclic, abelian non-cyclic groups, ${\mathcal A}_{4}$, ${\mathcal A}_{5}$ and ${\mathfrak S}_{4}$. For general $G$, we provide the value of $\sigma^{pnf}_{hyp}(G)$, correcting a previous statement on this value. In particular, we observe that every genus is the purely non-free genus of some group. As an application, we describe the minimal purely non-free actions for the family of metacyclic groups $G^{0}_{s}(n)=\langle x,y: x^{n}=y^{2}=1, yxy=x^{s}\rangle,$ where $n \geq 3$ and $2 \leq s \leq n-1$ is such that $s^{2} \equiv 1 \mod(n)$.

Earthquake catalogs underpin studies such as fault structure, earthquake processes and seismic hazard, and their value depends on being both complete and reliable. Machine learning has improved individual tasks of earthquake monitoring, but processing stations independently discards the consistency of arrivals across the network, and running tasks in sequence fragments the information describing each earthquake. We present EQNet2, a geometry-aware multi-station Transformer that jointly predicts phase arrivals, first-motion polarities, source location and magnitude, and the association of arrivals into events. On a Northern California test dataset and on continuous waveforms from Ridgecrest and the San Francisco Bay Area, EQNet2 recovers more weak arrivals, produces fewer false detections, and associates more picks into located events. Its learned attention links information across stations along time–distance patterns consistent with seismic moveout. As seismic networks grow denser, exploiting multi-station coherence and event-centric prediction offers a path to more complete and reliable catalogs.

Bridge sampling estimates of marginal likelihoods are routinely reported with a delta-method Monte Carlo standard error (MCSE). Existing theory is asymptotic and cannot say whether a given run is long enough to trust the estimate or its MCSE. We give finite-sample answers. The optimal bridge estimator fits a logistic regression that separates posterior from proposal draws, and its accuracy is governed by their overlap, within a factor of two of the classification's Bayes error. Because the estimating function is a sum of bounded terms, no moment conditions are needed, and all bounds hold uniformly over targets, proposals and dimensions. A number of draws inversely proportional to the overlap suffices and is necessary in the worst case, paralleling the exponential-in-Kullback–Leibler requirement of importance sampling. A Berry–Esseen bound with an explicit constant holds at the optimal rate, and the MCSE is accurate once the effective overlap is large; it is estimated at no cost. For Markov chain draws, the required sample size grows in proportion to the sampler's relaxation time, unavoidably in the worst case. The estimated bridge, the fixed-point iteration and the fitted proposal matter only at second order, and we give free diagnostics for sampler failure and misestimated autocorrelation.

Understanding a narrative means knowing how every event relates in time to every other, yet large language models (LLMs) are mostly tested on isolated event pairs or coarse labels. We ask five LLMs to reconstruct the complete network of Allen's 13 interval relations for 360 English narratives whose events are placed relative to one another rather than by dates, and evaluate the outputs under a pre-registered protocol for agreement with our gold annotations, converse consistency and logical satisfiability. End to end, the best model clearly beats trivial baselines on macro-F1, our primary metric, yet under our pre-registered parser no model matches the accuracy of always answering "before", which the skewed labels inflate. Most of that accuracy gap is a parsing artifact: formatting errors in large matrices cost whole rows, and a post hoc parser that recovers the determined cells of malformed rows raises coverage from 0.60 to 0.84 and closes the gap to the baseline for the best model. On the cells they do answer, models recover precedence and containment but rarely overlap, and they fail most on flashbacks, where both the narration and the event list present the later event first. Logical consistency varies widely yet says little about agreement with gold: enforcing it raises accuracy mainly by filling empty cells, and is beaten there by the trivial control of guessing the majority label. Reliable structured output and robust handling of relative time are prerequisites for temporal reasoning over whole narratives.

Incomplete field-of-view (FOV) diffusion MRI (dMRI) acquisitions compromise whole-brain white matter analysis by excluding brain regions from tractography and connectivity studies. While deep generative models for dMRI FOV extension have been proposed, their generalizability remains limited by the acquisition-dependent nature of dMRI, where signal intensity varies with b-value, gradient direction, and scanner protocol, and the inherent 4D complexity of the data, making it difficult to train models that transfer across sites or acquisition schemes without retraining. We propose a flow-matching generative model that extends the dMRI FOV in both the superior and inferior directions by predicting apparent diffusion coefficients (ADC) in six directions, conditioned on the two nearest acquired slices and the full-FOV T1w image to enforce both local diffusion continuity and anatomical consistency. The ADC representation decouples the model from acquisition-specific b-values and gradient directions, providing a middle-ground across protocols. The model was trained on 3,103 subjects (5,105 sessions) across five different datasets spanning diverse acquisition schemes (b = 500 and 1000 s/mm$^2$; 21–192 gradient directions), four single-shell (b = 1000) and one multi-shell (b = 500/1000). Evaluated on 299 subjects across five independent datasets spanning b = 750–1500 s/mm$^2$ and 32–90 gradient directions, with single- and multi-shell schemes unseen during training, the proposed method achieves significantly improved bundle Dice score on downstream tractography (p < 0.01).

We prove the existence of Steiner minimal trees spanning a finite family of metric spaces in two cases of the Gromov–Hausdorff theory. In the compact Gromov–Hausdorff space $\mathcal{M}$, every finite family of nonempty compact metric spaces is spanned by a shortest Steiner minimal tree whose Steiner vertices are again nonempty compact metric spaces. In a fixed Gromov–Hausdorff cloud $\mathcal{C}$, every finite family of proper, isometrically cobounded spaces admits a finite-length Steiner minimal tree all of whose vertices remain in $\mathcal{C}$. The compact case uses compact replacement through distance profiles and the Hausdorff hyperspace, and the cloud case uses pointed ultralimits.

Continuous monitoring of generative AI systems requires detecting distribution shifts in the unstructured data they ingest and generate. Existing monitoring methods typically control false alarms by requiring the discrepancy measure used to detect shifts to be *pre-specified*. However, for unstructured data, pre-specification creates a sensitivity–breadth trade-off: a narrowly defined measure is sensitive to the anticipated shift but may miss unexpected changes, whereas a broadly defined measure may detect a wider range of shifts but with reduced sensitivity. Although adaptively hypothesizing the shift direction can overcome this limitation, such data-dependent adaptation generally invalidates conventional false-alarm guarantees. We reconcile adaptivity with statistical validity through testing-by-betting, reframing sequential changepoint detection as a betting game in which the bettor continually updates its hypothesized shift direction, or witness function, as new observations arrive. We introduce *asymptotic e-detectors*, which approximate the pre-change distribution using a calibration set of size $m$, and prove that they asymptotically control the average run length to a false alarm as $m$ increases. Empirically, calibration sets as small as $m=30$ provide effective control. We then develop adaptive monitoring strategies ranging from black-box bettors that learn from embeddings to interpretable bettors based on LLM-proposed concepts. Across three real-world datasets, including data from a clinical AI scribe deployment, adaptive monitors detect shifts faster than fixed monitors at the same false-alarm level. Further, when the true shifts are sparse and interpretable, LLM-concept monitors provide the fastest and most interpretable detection, while also guiding root-cause diagnosis for complex shifts.

This paper concerns the Helmholtz scattering problem for open waveguide networks, extending the analysis from [arXiv:2302.04353, arXiv:2310.05816, arXiv:2401.04674] to plane-wave incident data. We first reduce the problem to a Fredholm second-kind integral equation on an infinite interface. This produces an integral representation of the solution to the original scattering problem, which is then used to obtain detailed asymptotic estimates of the solution at infinity. To leading order and away from the waveguides, the scattered field is itself a plane wave. Its amplitude and phase shift exhibit a jump across a critical polar angle depending on the direction of the incident plane-wave. Our estimates also imply that the scattered field is outgoing in a weak sense, allowing us to establish a uniqueness result. We conclude by calculating the waveguide-mode contributions to the scattered field in a perturbative regime.

On-policy distillation (OPD) is a post-training approach for large language models (LLMs) that enables a student model to learn a teacher model's distribution from sequences sampled from the student's own policy. Despite its promising capability, OPD for diffusion language models (DLMs) remains underexplored and existing studies primarily focuses on DLM-to-DLM distillation. The limited availability of strong teacher DLMs and the flexible generation order of DLMs impose additional technical challenges as well. However, it remains unclear how to effectively distill knowledge from a strong autoregressive language model (ARM) into a weaker DLM, given the mismatch between the left-to-right decoding of ARMs and the flexible decoding order of DLMs. Inspired by Ni et al. (2026), we propose JustOPD, an ARM-to-DLM on-policy distillation method that uses autoregressive rollouts during training to align the student's generation process with the ARM teacher's left-to-right decoding. This alignment enables effective distillation from ARM teachers while preserving the DLM's flexible generation-order capability at inference time. Experiments using Qwen3-14B as the teacher and Dream-7B as the student demonstrate that JustOPD consistently improves mathematical reasoning performance over the base DLM across multiple divergence choices.

Cortisol is an essential hormone secreted by the adrenal glands. Abnormally low level of cortisol is referred to as adrenal insufficiency. This paper focuses on devising treatment strategies for patients suffering from adrenal insufficiency using a model-based approach. Two conditions of adrenal insufficiency are analysed. They are hypothalamic-pituitary (HP) and adrenal gland disorders. The hypothalamus, pituitary, and adrenal glands constitute the hypothalamus-pituitary-adrenal (HPA) axis, which regulates stress and maintains physiological homeostasis through cortisol secretion. A comparative analysis of the existing mathematical models of the HPA axis is conducted. Based on this analysis, a novel model of the HPA axis is proposed that incorporates endogenous circadian rhythm. This proposed model is used to design effective cortisol replacement strategies. In addition, hydrocortisone replacement therapies are proposed and validated with clinical data from the literature. The results presented in this paper are applicable to patient-specific treatment strategies for individuals with adrenal insufficiency.

Parametrized gates in a variational quantum eigensolver (VQE) live on a product of unitary groups, which makes Riemannian optimization a natural candidate for improving training. We test this candidate in the clean per-gate setting: each single-qubit gate is optimized directly on U (2) and compared with Adam on an unconstrained Euler-angle chart representing the same physical gate, initialized from the same unitaries and tuned with a separate learning-rate sweep. On a validated hydrogen-chain benchmark, we find no measurable practical advantage from the Riemannian machinery. The direct manifold optimizer and the Euler-coordinate baseline are interleaved in final error and success rate; the classical coordinate baseline is not slower in iterations on the successful cells; and the choices of Riemannian metric, retraction, and momentum vector transport do not change the conclusion. We also identify four failure modes that can manufacture an apparent positive result: a trivial product-state benchmark, a projected-matrix Adam baseline that discards most of its own step, a non-unitary inherited Cayley retraction that lets the reported energy pass below the variational minimum, and a learning-rate grid whose optimum sits at its boundary. Each failure mode produced a plausible multi-seed result in our own workflow before being caught. The practical output of this paper is therefore a controlled null result and a validation checklist for VQE optimizer comparisons.

The independence complex of the Kneser graph $\KG(n,k)$ is the simplicial complex of pairwise intersecting families of $n$-subsets of $[2n+k]$; equivalently, it is a Vietoris–Rips complex of the $n$-subsets of $[2n+k]$ under the symmetric-difference metric. Feng and Wang asked whether, for $n=3$, its reduced homology is nontrivial only in dimensions $6$ and $9$, and conjectured the rank in dimension $9$. We answer the question affirmatively over the rationals and determine all the associated Betti numbers. For every $k\ge0$, the only nonzero reduced Betti numbers using rationals of $\Ind(\KG(3,k))$ are \[ \beta_6=\binom{k+5}{7}+29\binom{k+6}{7} \qquad\text{and}\qquad \beta_9=\binom{k+6}{6}. \] The proof uses an inductive decomposition based on a simplicial retraction and Barmak's star clusters, Dowker duality together with Jonsson's computation of the homology of complexes of graphs with bounded covering number, and a relative Mayer–Vietoris spectral sequence which reduces the problem to the six-element sets.

The theory of matrix monotone functions of finite degree depends subtly on the matrix size. Motivated by this phenomenon, we introduce radial convexity of metrics with respect to matrix orders on the cone of positive definite matrices. We establish radial convexity of several natural metrics with respect to the Loewner order and then focus on the Frobenius norm. Using the Daleckii-Krein formula, we relate radial convexity to the positivity of the Hadamard square of the corresponding Loewner matrix. This leads us to conjecture that radial convexity of order $n$ is equivalent to matrix monotonicity of degree $n$, and we prove this equivalence in degree $2$. We further provide evidence in higher dimensions, obtain a characterization in degree $3$, and discuss extensions of our results to arbitrary unitarily invariant norms.

We present a method for enforcing exactly Dirichlet, Neumann, and Robin conditions on general triangular domains with arbitrarily curved boundaries. The algorithmic constructions developed here complements those in a previous work for general quadrilateral domains. Our method combines (i) the exact mapping between a general curved triangular domain and the standard triangle, (ii) TFC constrained expressions, and (iii) Barnhill transfinite interpolations, to provide explicit closed-form trial ansatze that mathematically satisfy the imposed boundary conditions exactly. Corner compatibility induced by the boundary conditions is pivotal, and enforcing exactly the imposed boundary conditions hinges on the exact enforcement of the induced compatibility constraints. The corner compatibility associated with the acute vertices of the standard triangle and the coupling between gradient components when the hypotenuse corresponds to a Neumann/Robin boundary pose particular challenges. Two strategies are critical to resolving these issues, by reformulating conditional corner constraints into equivalent unconditional forms and by introducing a separate TFC-style auxiliary edge construction together with an associated linear solve on the hypotenuse. We analyze the corner compatibility constraints and present formulations for the general form of functions that exactly satisfy the imposed Dirichlet, Neumann, or Robin conditions on curved triangular domains. The proposed method has been implemented using extreme learning machine type randomized neural networks for solving PDE problems. We present ample numerical experiments with linear/nonlinear PDE problems on a variety of domains with complex boundary geometries. Numerical results show that the current method has enforced the Dirichlet, Neumann, and Robin conditions on curved domain boundaries to the machine precision.

Amazing breakthroughs in reinforcement learning have taken place. Computers teach themselves to play Chess and Go and beat world champions. There is talk about expanding application areas towards general artificial intelligence (AI). The breakthroughs in Backgammon, Checkers, Chess, Atari, Go, Poker, and StarCraft have shown that we can build machines that exhibit intelligent game playing of the highest level. These successes have been widely publicized in the media, and inspire AI entrepreneurs and scientists alike. Reinforcement learning in games has become a mainstream AI research topic. It is a broad topic, and the successes build on a range of diverse techniques, from exact planning algorithms, to adaptive sampling, deep function approximation, and ingenious self-play methods. Perhaps because of the breadth of these technologies, or because of the recency of the breakthroughs, there are few books that explain these methods in depth. This book covers all methods in one comprehensive volume, explaining the latest research, bringing you as close as possible to working implementations, with many references to the original research papers. The programming examples in this book are in Python, the language in which most current reinforcement learning research is conducted. We help you to get started with machine learning frameworks such as Gym, TensorFlow, and Keras, and provide exercises to help understand how AI is learning to play.