Papers

Recursive self-improvement (RSI) in scientific research aims to improve agents' research capabilities through their own experimental experience. Scientific agent swarms can combine evidence and implementations from different research directions to guide their agents' subsequent experiments. However, poor experimental results can leave unclear whether to revise a hypothesis or its implementation, while overall program scores can hide components useful to other agents. To address these challenges, we introduce SwarmRSI, a framework that evolves scientific agent swarms for recursive self-improvement. We devise Feedback-Guided Swarm Mutation to refine hypotheses and their implementations using experimental feedback from other agents. We also propose Complementary Swarm Recombination to reuse retained hypotheses and complementary program components across agents, allowing partial results to support new research. Recursive Swarm Evolution integrates both mechanisms into a feedback loop, using new experimental evidence to guide subsequent research. Experiments on OpenProblems, NatureBench and MLE-Bench Lite demonstrate SwarmRSI's state-of-the-art (SOTA) performance. Further evaluations show that accumulated research experience improves performance on held-out tasks. Biologists also reproduce its programs for follow-up biological analyses.

We investigate the notion of holomorphic $p$-contact structure, that we introduced recently in the scalar case with H. Kasuya and L. Ugarte and in the line-bundle case with K. Broder, in the context of a holomorphic family of compact complex manifolds equipped with a holomorphic family of holomorphic line bundles. Firstly, we show that the existence of a holomorphic $p$-contact structure is deformation-open when a non-jumping assumption is made on the $\bar\partial$-cohomology of bidegree $(p,\,0)$. This ought to become a source of further examples of holomorphic $p$-contact manifolds. Secondly, we obtain a Berndtsson-type (semi-)positivity result for the holomorphic vector bundle over the base induced, as a direct image, by the line bundle on the total space that supports the p-contact structures. Thirdly, we introduce a moduli space of relative holomorphic p-contact structures and we show that it is Hausdorff and open in a certain quotient topological space.

Though modern AI models like LLMs predict discrete tokens, today's robotic foundation models represent actions as continuous vectors. In this paper, we challenge this common consensus by asking if continuous action parametrizations are superior to their discrete counterparts. We conduct an extensive comparison of single-step regression and multi-step flow variants of both continuous and discrete policies in both from-scratch single-task and multi-task VLA training, in both simulation and on real hardware. In all regimes, we find that discrete action prediction, even the most naive, uniform-bin quantization, can match or even outperform continuous-action policies in task success rate. Moreover, we find that the challenge to discrete action policies is not action precision, but rather factorization error introduced by product-distribution predictions across time and action dimensions. We show that, when paired with the correct interventions—iterative computation through discrete flows, or distribution sharpening through classifier-free guidance—discrete flow policies can turn this weakness into a strength, enabling more responsive action changes during a predicted action chunk. Lastly, we show that attempts to explicitly inject geometric information into discrete policies, e.g. through corruption processes that emulate continuous flow matching, do not yield consistent improvements. Taken together, our findings question existing assumptions that robotic action prediction requires distinct design decisions compared to other AI domains, and motivates the possibility of discrete action prediction as a compelling alternative to today's continuous action parametrizations. Link for the webpage \href{https://iscoyizj.github.io/discrete-action-website/}{https://iscoyizj.github.io/discrete-action-website/}

The category of all topological spaces and homotopy classes of maps is cartesian closed. In fact, the $(\infty, 1)$-category of all topological spaces and continuous maps, localised at homotopy equivalences, is cartesian closed. We model the exponential object using a topological space whose points are continuous maps equipped with putative neighbourhood filters. In general, this topological space will not even be $T_1$-separated, but the evaluation map induces a natural weak homotopy equivalence witnessing the universal property.

Detecting unseen malware variants remains a major challenge as malware families continuously evolve. This issue is exacerbated in data-scarce scenarios such as emerging malware campaigns, where only a few samples are available to train robust detectors. A common strategy is to proactively generate malware variants and use them as augmentation data to expose models to a broader range of potential threats. Existing approaches that generate functional malware variants through adversarial manipulation suffer from several important limitations. Generated samples must remain executable and preserve their malicious behavior, requiring costly validation through execution and analysis. Moreover, the adversarial objective provides limited control over the resulting augmentation dataset: it cannot guarantee the generation of sufficiently diverse variants, or even a single successful sample, making the effectiveness of the augmentation process inherently unpredictable. Finally, these approaches typically rely on a well-trained detector to guide generation, an assumption that may not hold in data-scarce scenarios such as early-stage malware campaigns. We formally and empirically demonstrate these limitations and propose Phantomware, a data augmentation framework designed for extreme data scarce scenarios. Phantomware operates entirely in the representation space of a pretrained malware detector, avoiding the costly generation and validation of functional binaries. By exploiting latent-space geometry, it generates synthetic malware representations of a target family in a controlled manner, providing a constructive guarantee on the generation of diverse training samples. Extensive experiments on real malware families demonstrate that Phantomware improves detection of unseen variants and provides robustness against both real and adversarially generated malware samples.

This paper considers reliable in-sequence molecular communication, where bio-nanomachines exchange molecule-encoded information in a specified order, and investigates a new simulation-based method to incrementally evaluate communication protocols by gradually increasing the complexity of experimental settings. The proposed method, called Affine Memory Replay (AMR), builds a continual learning (CL) model to estimate communication performance between bio-nanomachines. AMR stores past simulation samples and synthesizes additional replay samples through interpolation and extrapolation between past samples. Through incremental performance estimation tasks, it replays both past and synthetic samples to regularize model updates, helping the model adapt to new experimental settings while retaining knowledge from earlier ones. Experimental results show that AMR significantly outperforms existing CL methods in maintaining overall estimation accuracy across sequentially learned estimation tasks. It effectively balances memory stability and learning plasticity under moderate training cost.

We present new superdark calibration files (DARKFILEs) and a new dark correction table (TDCTAB) for use with data taken by the Space Telescope Imaging Spectrograph near-ultraviolet microchannel array (STIS NUV-MAMA) detector for HST Cycle 19 onward. These superdarks include 12 years of NUV dark exposures and make use of an updated temperature scaling equation developed in 2013 to more accurately characterize the dark rate. In tested datasets, the new superdarks provide an average 3% increase in SNR and smooth out a ring artifact left previously by oversubtraction of the dark rate. The new superdarks and TDCTAB have been delivered to the Calibration Reference Data System (CRDS) and have been used to recalibrate all affected datasets in the Mikulski Archive for Space Telescopes (MAST).

Existing long-horizon agent benchmarks often emphasize final task success while providing limited assessment of intermediate execution validity. Yet real-world business operations require agents to navigate extensive, potentially outdated or conflicting policies while satisfying critical constraints throughout execution. These requirements are particularly consequential in high-stakes domains such as payments, where intermediate violations can incur financial losses or regulatory consequences regardless of the final outcome. To bridge this evaluation gap, we introduce BizBench, a multi-turn benchmark grounded in real-world procedures from a leading payments company. Its executable environment integrates 789 knowledge documents and 601 discoverable business tools, with consistent policies, states, and tool behavior. The benchmark comprises 50 tasks involving resource constraints, temporal dependencies, prior commitments, and conflicting or outdated guidance. Each task is validated through reference execution, while deterministic checks assess both goal attainment and execution validity without prescribing a single solution path. Experiments reveal substantial limitations even among frontier models: Claude Opus 5 and GPT-6 Sol achieve strict success rates of only 64.0% and 61.3%, respectively. For Qwen3.8-27B, goal attainment reaches 42.0%, but strict success drops to 32.7%, demonstrating how outcome-based evaluation can mask invalid execution. Beyond success rates, DeepSeek-V4.1-Flash averages 191.3 tool calls per run, approximately 70% more than reference executions, revealing substantial execution overhead. Overall, BizBench establishes a testbed for evaluating and advancing long-horizon agents beyond task completion, toward reliable and constraint-compliant execution in complex business environments.

The matter inside a spinning-down neutron star (NS) undergoes a slow compression that may drive it out of chemical equilibrium and, consequently, trigger non-equilibrium nuclear reactions that can heat up the star. This phenomenon is called rotochemical heating, and it is relatively well-investigated in the literature. Historically, it has been studied almost exclusively via 1D spherically symmetric models. However, rapidly rotating NSs are not spherically symmetric. We present pioneering simulations of rotochemical heating in full General Relativity in 2D axially symmetric models without relying on the slow rotation approximation. We explicitly follow thermal, mechanical, and chemical evolution of rotating NSs as they spin down taking into account both radial and angular distributions of temperature, composition, and mechanical structure. The simplest model is employed: only nucleonic composition, only the non-equilibrium modified Urca process as the main heating/cooling mechanism, and no superfluidity. We analyze the impact of the magnetic field strength as well as the effects of the heat blanketing envelope, initial rotation frequency, NS mass, and core EOS. Most of our conclusions coincide with the results of previous works on this topic, with one important exception. We have found a strong dependence of the cooling curves during the rotochemical heating phases on the core EOS, whereas in the literature it was regarded as very weak. The most likely explanation for this discrepancy is that the authors of the previous works have not considered EOSs with wide enough range of variation of the symmetry energy as a function of density.

While sensitivity analysis for linear programs with parametric right- hand side vectors is well established, cases involving parametric constraint matrices remain less explored. To address this gap, we investigate linear programs where the constraint matrix depends linearly on a scalar parameter. Our goal is to compute the minimum and maximum optimal values of these parametric linear programs within a fixed parameter range. This corresponds to solving two bilevel programs. We show that both decision versions of these optimization problems are NP-complete by proving both NP-hardness and membership in NP. We additionally identify three sufficient conditions on the parametric matrix under which the bilevel programs are no longer NP-hard. We introduce a reformulation for both problems, together with a Farkas-based certificate for the lower-level feasibility condition, and compare the execution time for solving them on standard benchmark instances. The experiments highlight a practical dichotomy. Although both bilevel optimizations are NP-hard, computing the parameter value corresponding to the minimum of the linear program is fast, whereas computing the value corresponding to the maximum is not. We relate this asymmetry to the parametric equality constraints, whose free dual variables leave the relaxation of the maximum reformulation unbounded.

We study the parabolic Anderson model $\partial_t u=\Delta u+\beta u\,\dot W_\alpha$ on an unbounded Ahlfors $d_h$-regular Dirichlet metric measure space whose heat kernel satisfies two-sided sub-Gaussian estimates with walk dimension $d_w$. The noise is white in time and, in space, either white ($\alpha=0$) or colored through the Riesz kernel of $(-\Delta)^{-2\alpha}$, where $0\le 2\alpha<d_h/d_w<1+2\alpha$. No recurrence is assumed. We prove existence, uniqueness and Hölder regularity of the mild Skorohod solution for bounded initial conditions. For nonzero nonnegative bounded initial conditions, the solution is positive and its moment Lyapunov exponents exist for every real $p\ge1$, independently of the spatial point and the initial condition. For $p\ge2$, these exponents have order $p\,(\beta^2(p-1))^{d_w/((1+2\alpha)d_w-d_h)}$, with upper and lower bounds matching up to constants.

We develop a dual-inertial framework for convex optimization that introduces inertia in dual coordinates while retaining the Bregman proximal-type oracles. For unrestarted iterations, uniformly bounded positive directional variation gives a last-iterate $\cO((1+\log k)/k^2)$ rate, improving to $\cO(k^{-2})$ under additional weighted summability. A computable safeguarded algorithm gives a global $\cO(N^{-1})$ objective-gap bound for the best accepted point, where $N$ counts all proximal oracle calls. Under local strong-convexity and kernel regularity conditions, all three safeguarded methods restart finitely and attain a final-epoch last-iterate $\cO(N^{-2})$ rate. Numerical experiments compare objective accuracy and computational cost on entropy regression and nonnegative least squares. Comparative numerical experiments illustrate the performance advantages of the proposed methods.

Retrieval-Augmented Generation (RAG) models have become almost a cornerstone of open-domain question answering tasks; in which the quality of answers generated in the process, greatly relies on the effectiveness of retrieval mechanisms. In the realm of LLM, the last two years have seen tremendous progress in RAG architectures, with many models reporting to have improved performance through various design innovations. The cultural question and answering systems of South Asia, in particular, present a specific challenge since the region is home to the vast number of languages and traditions, which are closely interrelated. Along with these deeply intertwined historical narratives increase the need for culturally grounded and context-aware interpretation. Despite major architectural advances in LLMs, claiming significant improvements, challenges remain when answering these region specific questions in native languages. In this study, we aim to extract high-performing factors to guide the development of optimized hybrid RAG systems. We further explore whether the integration of RAG changes performance differently for culturally grounded and non-cultural questions in a South Asian multilingual setup, and whether RAG produces a higher overall percentage of correct answers than contexted answering. We also examine how LLM performance changes when RAG methods are integrated into the question-answering pipeline in comparison with non-contexted and contexted answering and, evaluate whether the introduction of RAG benefits popular or non-popular culturally grounded questions more. Not only does our work evaluate the current methods but also carves a path for future designs of more accurate and efficient question-answering pipelines.

Tabular foundation models enable in-context learning by predicting unseen observations from a labeled support table without task-specific parameter updates. Despite recent advances, most existing approaches rely on dense architectures that apply the same transformations across features and datasets. We introduce OrionMoE-Small, a compact tabular foundation model that combines distribution-aware representations with dataset-conditioned sparse mixture-of-experts computation. Built on a dual-axis Transformer architecture, OrionMoE-Small selectively activates experts along the feature axis, with routing decisions informed by both individual feature representations and the broader dataset context. This design enables adaptive allocation of computation across heterogeneous tabular problems while retaining a shared computational pathway across tasks. OrionMoE-Small is pretrained entirely on synthetic classification tasks generated from a structured causal prior and performs downstream prediction without gradient-based adaptation. Experiments across multiple tabular benchmarks demonstrate competitive performance against substantially larger foundation models, with particularly strong results on small-context and high-dimensional datasets. Further analysis reveals favorable relative performance under class imbalance and highlights the importance of feature composition.

Audits of large language model (LLM) extraction from full-length documents report severe unreliability, leaving two questions open: what the failures actually consist of, and whether confidence signals can turn unreliable output into usable output. We run a preregistered, fully frozen evaluation of two new-generation low-cost LLMs (DeepSeek-V4-Flash, GLM-5.3-Flash) with a flagship-tier control (GPT-6.1-Sol) on a public corpus of 354 incident campaigns (315 unique reports, 630 model-report evaluations), scoring extraction quality against gold, confidence calibration, and robustness to single-sentence perturbations. Three findings emerge. (1) Unreliability persists across tiers: micro precision is 0.03-0.34 and verbalized calibration error (ECE) 0.53-0.94 across both splits; the gap does not close at the flagship tier. (2) The dominant failure is not hallucination: a full mention census plus a three-tier annotation of 360 sampled candidates (two independent LLM annotators from vendors outside the evaluated set, Cohen's kappa = 0.79 on the preregistered acceptance line, fully disclosed) back-projects to 65.6% background mentions and 19.9% gold-scope gaps, with only 1.2% unsupported candidates. (3) Confidence ranks but does not verify: zero-cost self-consistency-frequency tiering passes a preregistered top-30% precision-gain bar in all six test-split model-type cells (4/6 on train), the zero-cost text-frequency x SC frequency combination wins 3, ties 2, loses 1 across cells, yet relation-level counterfactuals on all 45 targets show that a single misattribution sentence flips high-confidence judgments in 95.6-100% of cases, and even a background sentence explicitly marked as unrelated still binds a fake vulnerability in up to 26.7% of them. We conclude with a deployment boundary: confidence tiering may set review priorities, never truth.

For the fundamental problem of prediction from expert advice, Freund asked whether one could compete with a comparator that switches $k$ times among only $m$ experts. For a discrete set of experts, this problem was resolved by Bousquet and Warmuth (2020), who gave an efficient algorithm with optimal regret matching the information-theoretic limit. We study the analogous problem in online convex optimization, where the action space is continuous. An inefficient algorithm can be derived by discretizing the action space and applying the Bousquet–Warmuth algorithm, giving regret of $\widetilde{O}(md+k)$, while the best known efficient algorithm for this problem has regret $\widetilde{O}(dk)$. We show that this computational inefficiency is inherent, establishing a separation between the information-theoretic and polynomial-time regret achievable for this problem. Specifically, under the hypergraphic planted-clique conjecture, we show that for every polynomial-time algorithm there exist instances for which it incurs at least $(k\sqrt d)^{1-o(1)}$ regret.

Poor generalisation in deep neural networks is frequently associated with convergence to sharp minima. Sharpness-Aware Minimisation (SAM) improves generalisation of deep neural networks by minimising a combination of the training loss and the local sharpness of the loss landscape, biasing convergence towards flat minima. However, SAM relies on a fixed perturbation radius $\rho$ that ignores local curvature, leading to inefficient behaviour: the perturbation may be too small to escape poor basins defined by sharp local minima, or too large to allow convergence to solutions with good loss-sharpness tradeoff. We reinterpret $\rho$ as the intensity of the perturbation signal and propose Curvature-Aware Adaptive Perturbation for SAM (CAP-SAM), a geometry-aware method that modulates the perturbation radius based on a sharpness ratio, enabling a more effective search for robust minima on the loss landscape. We prove that CAP-SAM preserves the same convergence rate as standard SAM in non-convex stochastic settings. Extensive experiments indicate that our approach consistently outperforms SAM and its most prominent variants across diverse network architectures on CIFAR-10, CIFAR-100, and ImageNet.

Let $G$ be the terminal graph of the random triangle-removal process started from $K_n$. We prove that $e(G)/n^{3/2}\to1/(2\sqrt2)$ in $L^2$. More generally, we obtain the analogous result for any fixed strictly $k$-balanced $k$-graph. Our main tool is a more general theorem on random greedy matchings in nearly regular uniform hypergraphs with few short cycles under a stability assumption on the survival equations.

Momentum is a classical tool for accelerating convex optimization and a standard component of modern large language model training. Yet recent empirical work has questioned the usefulness of first-moment momentum in reinforcement learning with verifiable rewards (RLVR). We study this question through a bandit formulation of RLVR with bias-corrected exponential moving average (EMA) momentum, where the EMA coefficient $\beta$ controls how quickly the weights on past gradients decay. When policy gradients are evaluated exactly, we prove that increasing $\beta$ within a range that prevents overshoot strictly accelerates optimization. When gradients are estimated from sampled responses, a positive expected gain can be obscured by sampling fluctuations. We show that as the learning rate $\eta$ decreases, the expected gain with fixed $\beta$ vanishes faster than these fluctuations, whereas increasing $\beta$ to keep $\eta/(1-\beta)$ within the safe regime preserves a nonvanishing gain that eventually dominates them. Bandit simulations quantitatively validate these predictions. RLVR experiments on Qwen3-4B-Base using SGD with EMA and AdamW support our principle of choosing larger coefficients in practical LLM post-training. On Qwen2.5-32B with AdamW, simply increasing $\beta_1$ improves mean validation accuracy by nearly $2$ percentage points over the conventional choice, demonstrating the principle's effectiveness at a larger model scale.

To reach top accuracy, current fully supervised instance segmentation methods severely rely on large-scale pixel-wise labeled datasets. They are usually expensive and time-consuming to obtain. Though weakly or semi-supervised methods utilize cheap bounding box labeled, image-level labeled or unlabeled samples to save the labeling cost, their performance is largely sacrificed. To save labeling cost without losing much performance, in this paper, we present a pipeline that can utilize economical bounding box labels and accurate pixel-wise labels in a hybrid way. Specifically, we design two ancillary models to learn label noise suppression and obtain accurate pseudo pixel-wise labels from bounding box labels for training. One is designed to suppress mislabeling between foreground and background, and the other is designed to suppress noise from mislabeling of instances. Moreover, we exploit category-aware spatial attention module, category constraint module, instance constraint module, and self-learning training approach to improve the accuracy of pseudo labels. Experiments on the PASCAL VOC 2012 and the Cityscapes datasets show that our method can achieve competitive performance with much less labeling cost.

Standard competition-based multiple testing methods – such as the knockoff filter and the target-decoy procedure – are typically formulated as head-to-head competitions between a test unit and its assigned negative control unit. Motivated by real-world applications, which leverage any available negative controls, we study this competition-based framework in more intricate settings where competitions vary in size and structure. We propose new methods based on compound p-values – p-values that are superuniform only on aggregate – that enable powerful multiple testing with broader competition structures, via the Bonferroni and Benjamini-Hochberg procedures. The former provides exact familywise error rate control, whereas the latter may incur a false discovery rate inflation factor, for which we derive upper bounds across a range of settings. We showcase the performance of our general competition-based testing methodology through extensive simulations and real-data applications, including analyses of virtual spatial transcriptomics and CRISPR screen datasets that exhibit precisely the complex competition structures that motivate our methods.

The resetting of stochastic processes has become a key mechanism appearing in many research fields, such as searching for a target, the transmission control protocol, biology, chemistry, finance, insurance, etc., with the main interest concerning the first passage time (FPT) by the process under resetting. In this paper, we focus on the exit time $\mathcal{T}_L$ from the interval $(-L, L)$ of Lévy flights (LF) under resetting, but we also consider spectrally negative Lévy processes (SNLP). As the main result, we find the asymptotics of the mean first passage time (MFPT) as $L$ tends to $+ \infty$. For LF with $\alpha \in (0, 2)$, the scaling behaves as $L^\alpha$ regardless of the reset rate and the asymmetry of the initial condition, a behaviour completely different from that for the known case of Brownian Motion ($\alpha=2$). For the symmetric $\alpha$-stable case with $\alpha \in (0, 2)$, we also found that taking $r \to \infty$ with fixed $L$ yields a finite MFPT, which therefore may be used as a valid approximation. For completely asymmetric stable processes, we show that the scaling is very different from that for LF, and point out a method to obtain the exact distribution of FPT under resetting. The analytical calculations are verified by numerical simulations of the processes under resetting.

We develop a self-supervised reinforcement learning algorithm that learns to build towers of up to eight cubes from scratch, without prior data, rewards, demonstrations, or pretrained models. Our method, Contrastive Planning via Spherical Linear Interpolation (C-Slerp), uses contrastive representations as reusable building blocks and repeatedly composes them to infer a sequence of intermediate waypoints. Under our hyperspherical parameterization, the learned representations allow intermediate waypoints to be inferred in closed form through spherical linear interpolation, without training a separate high-level policy or performing graph search. Importantly, the representation geometry also tells us how to adjust the number of waypoints based on the difficulty of each goal-reaching problem. Experiments show that our method succeeds on long-horizon navigation and robotic manipulation tasks, including settings where strong baselines fail to achieve a single success. Ablations further show that using a fixed number of waypoints substantially degrades performance, highlighting the importance of adapting the plan to task difficulty.

A perfect 1-factorization of the complete graph $K_{2t}$ is a partition of its edges into perfect matchings such that the union of any two of them is a Hamilton cycle. Kotzig conjectured in 1963 that $K_{2t}$ has a perfect 1-factorization for every $t \ge 2$, but apart from some infinite families and a list of sporadic orders the conjecture is open, and $92$ was one of the nine orders up to $100$ for which no perfect 1-factorization was known. In this paper we construct two non-isomorphic perfect 1-factorizations of $K_{92}$.

For a graph $G$, let $\mathrm{tw}(G)$, $\mathrm{tree}\text{-}\alpha(G)$, and $\mathrm{tree}\text{-}\chi(G)$ denote its treewidth, tree-independence number, and tree-chromatic number, respectively. We prove that every nonempty graph $G$ satisfies $\mathrm{tw}(G)+1\leq\mathrm{tree}\text{-}\alpha(G)\mathrm{tree}\text{-}\chi(G)(1+\lfloor\log_2(\mathrm{tree}\text{-}\alpha(G))\rfloor)$. This improves the quadratic bound of Koutsoutis, Krause, Liu, Redzic, and Ueckerdt. For every pair of integers $n,k\geq2$, we construct a graph $G$ with $\mathrm{tree}\text{-}\alpha(G)=n$, $\mathrm{tree}\text{-}\chi(G)=k$, and $\mathrm{tw}(G)=\Omega(nk\log_2 n)$. Thus the upper bound is best possible up to a constant factor.

In this article, we provide a complete answer to one of the most classical conjectures in differential geometry (of foliations). More precisely, we present a complete positive answer to the classical conjecture, due to Meeks III, Pérez, and Ros (2008), that states that every foliation on $\mathbb{R}^{n+1}$ by (possibly varying) CMC hypersurfaces is a foliation by minimal hypersurfaces. Moreover, if $n\leq 6$, such a CMC foliation must consist of parallel hyperplanes.

Conventional deep convection schemes remain a primary source of uncertainty in global climate models, yet replacing them with data-driven alternatives while maintaining physical consistency and numerical stability is an open challenge. We present a hybrid configuration of the Community Atmosphere Model version 6 (CAM6), in which a fully connected deep neural network replaces the Zhang-McFarlane deep convection scheme (ZM), while radiation, turbulence, and other cloud processes are left unchanged. The network is trained on output from a storm-resolving global model - the global System for Atmospheric Modeling (gSAM) - using a coarse-graining approach to derive subgrid-scale convective fluxes and tendencies. This data-driven parameterization is ported to CAM6 and found to run stably for more than a decade while conserving energy and moisture. It produces a broadly realistic climatology without retuning the remaining parameterizations. Relative to the default Zhang-McFarlane scheme, the neural network parameterization substantially improves some long-standing model deficiencies. In particular, the biases in light precipitation and extreme precipitation are markedly reduced, and the diurnal cycle of tropical land precipitation shifts from a spurious midday peak over many regions to the observed late-afternoon peak. These results support targeted ML-based convection schemes as a viable path toward reducing systematic model errors in next-generation climate models.

Radical parametrizations describe algebraic varieties by means of nested radicals of polynomials, and their theory relies on three varieties associated with a parametrization, namely, the incidence, tower and radical varieties. In this paper we revisit this framework from a computational point of view and show that it is governed by the radical tower rather than by the particular parametrization. We prove that the tower variety depends only on the tower and that the incidence variety is always irreducible, so that no prime decomposition is needed. Moreover, we show how to replace a tower by an isomorphic one whose tower variety is defined by the equations of the tower, so that it can be computed without elimination. We introduce universal radical parametrizations and we use them to settle, under a natural hypothesis on the coefficients, an open case in the transformation of differential equations with radical coefficients into equations with rational coefficients. We show that radical subtowers form a lattice, which yields minimal radical parametrizations, and we give algorithms to compute them. We also prove that dominance between towers induces a unique dominant rational map between their tower varieties, through which unirationality is transferred. Finally, we study towers over subfields of $\mathbb{C}$ and, in the real case, relate the real points of the tower variety to the real domains of the associated parametrizations.

Most of qualitative properties of finite element (FE) approximations for various PDE models, in particular the validity of the discrete maximum principles (DMPs), are usually guaranteed by certain geometric properties (e.g. acuteness/nonobtuseness) of FE meshes used. In this work, we revisit the linear FE discretization of the Poisson equation and discuss, in the context of DMPs, the representation of the stiffness matrix entries in terms of the edge lengths and cotangents of the dihedral angles of the tetrahedral meshes, thus avoiding the common use of face areas and element volumes. Using this type of entry representation, certain structural properties of meshes, and simple trigonometric relations, as proposed in the two-dimensional case by Strang and Fix, we show that DMPs still hold (as the stiffness matrices remain $M$-matrices) for many tetrahedral configurations with obtuse dihedral angles. Such a relaxation of the standard acuteness/nonobtuseness conditions can be vital in practical calculations, as these angle properties are difficult to preserve in global and local mesh refinements. Several representative examples are presented.

Feldman's memorization lower bound assigns a frequency coefficient based on how often a point appears in the training sample. We give an exact three-point counterexample to its conditional and averaged forms, using independent fair labels and a permutation-equivariant learner. A sequence of sampling priors makes the ratio of the proposed penalty to actual excess risk grow exponentially, ruling out a universal positive constant-factor repair. The priors vary with sample size, the offending profiles become rare, and both quantities vanish. Retaining the original model, we obtain an exact replacement using full-sample posterior probabilities. Under exchangeability, these depend on multiplicity and the complete frequency profile. We characterize the largest multiplicity-only coefficients valid for unrestricted learners. The original coefficient has the posterior-mean form of Good's exact frequency formula and is an occupancy-weighted average of the corrected coefficients.