Accurate clinical diagnosis requires extensive domain knowledge and complex clinical reasoning capabilities. Although large language models (LLMs) hold great potential for clinical reasoning, their high computational and memory requirements limit their deployment in resource-constrained environments. Knowledge distillation (KD) can compress LLM capabilities into smaller models, but traditional KD merely transfers superficial answer patterns and fails to preserve the structured reasoning required for reliable diagnosis. To address this, we propose a two-stage distillation framework, MedThink, designed to cultivate robust clinical reasoning in small language models (SLMs). In the first stage, a teacher LLM screens data and injects domain-knowledge explanations to fine-tune a student model, establishing a knowledge foundation. In the second stage, the teacher evaluates the student's errors, generates reasoning chains linking knowledge to correct answers, and refines the student's diagnostic reasoning through a second round of fine-tuning. We evaluate MedThink on general medical benchmarks and a gastroenterology dataset comprising 955 question-answer pairs. Experiments demonstrate that MedThink outperforms six distillation strategies in all benchmarks: achieving an improvement of up to 12.7% over the student baseline in general tasks, and reaching a total top accuracy of 56.4% in gastroenterology evaluation. This indicates that iterative distillation centered on reasoning can significantly enhance the diagnostic accuracy and generalization capabilities of SLMs whilst maintaining computational efficiency. Our code and data are publicly available at https://github.com/destinybird/PrecisionBoost.

The use of chatbots for various forms of companionship is growing rapidly, raising a myriad of questions about simulated relationships, emotional dependence, and psychological harm. While major platforms such as ChatGPT, Grok, and Character.AI are the subject of a growing body of research and legal inquiries, apps explicitly built for simulating intimate interpersonal relationships remain under-explored. In this work, we evaluate the five most popular AI companion mobile applications in the EU and UK markets for factors that encourage parasocial interaction and may manipulate users. We do this by manually annotating the user experience each offers. Specifically, we systematically record and quantify design dark patterns, anthropomorphism, stereotypes, erotica, and technical performance issues. We find that all apps contain substantial dark patterns aimed at increasing monetisation and user engagement. Erotica and gamification features such as levelling are also prevalent, and although other features vary considerably between applications, all apps have highly anthropomorphic design. These findings shed light on the mechanics used to leverage users' simulated relationships. On that basis, we put forward concrete recommendations for regulators to strengthen consumer protection in this rapidly emerging market. Content warning: This article contains objectifying images of women, erotic images, textual references to incest, and other potentially sensitive, offensive, and distressing text.

Education is one of the most promising real-world applications for Large Language Models (LLMs). However, current LLMs rely on static pre-training knowledge and lack adaptation to individual learners, while existing RAG systems fall short in delivering personalized, guided feedback. To bridge this gap, we present DeepTutor, a fully open-source agentic framework that unifies citation-grounded problem tutoring with difficulty-calibrated question generation. A hybrid personalization engine couples static knowledge grounding with dynamic learner memory, continuously adapting each interaction to the student's evolving needs. The same personalization substrate further extends to adaptive learning workflows, interactive books, and proactive multi-channel tutoring agents. To evaluate personalized tutoring, we introduce TutorBench, an interactive benchmark incorporating customized learner profiles grounded in university-level curricula across five domains. We further propose an LLM-based first-person interactive evaluation protocol that conducts assessments via a profile-driven student simulator. Complementary evaluations on established benchmarks, supported by human-alignment and ablation studies, confirm the framework's robustness and general utility. Results show that DeepTutor improves personalized metrics by 10.8\% on average and strengthens general agentic reasoning across five backbone models by 29.4\%.

Distillation via sampling reasoning traces exposes closed-source frontier models to adversarial third parties who can bypass their guardrails and misappropriate their capabilities. Antidistillation methods aim to address this by poisoning reasoning traces to hinder student model learning while preserving teacher performance. However, current methods overlook detectability, both semantic and syntactic, which erodes trust in the teacher's outputs and signals the defense's presence to adversaries. We address this gap by formulating antidistillation as a Stackelberg game whose constraint set explicitly encodes detectability, and show that perturbing sparingly offers an effective, less detectable alternative to poisoning the full trace. Drawing on mechanistic interpretability, we identify thought anchors, sentences with disproportionate counterfactual influence on model outputs, as a principled sparse target: critical to reasoning yet minimally detectable. We instantiate this in TraceGuard, a training-free, black-box proof-of-concept that locates thought anchors via branching-token detection and poisons them to degrade student distillation while preserving trace coherence.

Agentic computing systems, while immensely capable, raise serious security, privacy, and safety concerns. A key issue is that the full set of functionalities offered by these systems, combined with their probabilistic execution flows, is not known beforehand. Given this lack of characterization, it is challenging to validate whether a system has successfully carried out the user's intended task or instead executed irrelevant actions, potentially as a consequence of compromise. We present \emph{Agent Sentry}, a runtime defense that learns a bound on an agent's benign execution from prior legitimate executions and flags any action that falls outside this bound. Agent Sentry layers three complementary checks: a structural classifier over the sequence of actions and the provenance of each function's arguments; a deterministic allowlist check over sensitive argument values; and an LLM judge, invoked only on the residual of actions where the first two checks cannot safely decide between a legitimate new request and a carefully crafted injection. We demonstrate the effectiveness of Agent Sentry in AgentDojo and AgentDyn by blocking 94.3\% of successful injections while allowing 95.1\% of benign executions, without modifying the agent, its tools, or the LLM.

Security policy enforcement in contemporary agentic systems predominantly consists of embedding natural-language policies within an agent's system prompt and delegating compliance to the agent's reasoning. This approach admits no formal enforcement guarantee and cannot express policies whose satisfaction depends on the causal history of an execution, a gap that becomes acute in multi-agent systems, where enforcement must reason across agents. We argue that policy enforcement in agentic systems is most naturally understood as a cross-cutting concern, and propose a framework grounded in aspect-oriented programming that specifies policies independent of the agent's reasoning and enforces them at every policy-relevant decision. Policies are written in Datalog over a set of abstract predicates describing the execution context, an observability service governed by a formal assume/guarantee contract maintains these predicates, and a reference monitor consults the policy at each action to produce an enforcement decision. When the environment contract holds, enforcement decisions coincide with the policy's intended semantics. We adopt Datalog as the policy language, a natural fit because it supports declarative rule specification, admits recursion for policies over transitive relationships, and yields deterministic enforcement. Datalog further admits tractable static analyses for contradiction, redundancy, subsumption, and conditional reachability, enabling authors to verify policy intent and surface ambiguities inherent in natural-language specifications. We realize the framework in FORGE, which enforces policies over agentic deployments without modification to the underlying agents. We evaluate FORGE on three case studies: information flow policies for prompt injection defense, approval workflows in a multi-agent pharmacovigilance system, and organizational policies for customer service.

A central challenge in developing Multimodal Large Language Models (MLLMs) is effectively integrating heterogeneous inputs into a cohesive reasoning engine. Current paradigms predominantly rely on modular architectures that introduce modality-specific encoders and cross-modal fusion mechanisms. However, these designs are fundamentally bottlenecked by a geometric modality gap, forcing the LLM to expend significant computational capacity on geometric reconciliation rather than deep cross-modal reasoning. In this work, we formally characterize this modality gap and theoretically demonstrate that native architectures, specifically those employing a unified vocabulary, intrinsically maintain a zero-gap state across all hidden layers. Guided by these theoretical findings, we propose \textit{One Tokenizer}, a native architecture that maps all modalities directly into a shared token space. We empirically validate this framework on a DNA--text multimodal testbed. Our extensive evaluations reveal that by achieving seamless integration within the LLM's native latent space, One Tokenizer consistently outperforms encoder-based modular counterparts, providing a fundamentally superior framework for deep biological reasoning.

Combinatorial optimization problems are central to both practical applications and the development of optimization methods. While classical and quantum algorithms have been refined over decades, machine learning--assisted approaches are comparatively recent and have not yet consistently outperformed simple, state-of-the-art classical methods. Here, we focus on a class of Quadratic Unconstrained Binary Optimization (QUBO) problems, specifically the challenge of finding minimum energy configurations in three-dimensional Ising spin glasses. We use a Global Annealing Monte Carlo algorithm that integrates standard local moves with global moves proposed via machine learning. We show that local moves play a crucial role in achieving optimal performance. Benchmarking against Simulated Annealing and Population Annealing, we demonstrate that Global Annealing not only surpasses the performance of Simulated Annealing but also exhibits greater robustness than Population Annealing, maintaining effectiveness across problem hardness and system size without hyperparameter tuning. These results provide clear and robust evidence that a machine learning--assisted optimization method can exceed the capabilities of classical state-of-the-art techniques in a combinatorial optimization setting.

Malware proliferation is increasing at a tremendous rate, with hundreds of thousands of new samples identified daily. Manual investigation of such a vast amount of malware is an unrealistic, time-consuming, and overwhelming task. To cope with this volume, there is a clear need to develop specialized techniques and efficient tools for preliminary filtering that can group malware based on semantic similarity. In this paper, we propose TrapNet, a novel, scalable, and unsupervised framework for malware fingerprinting and grouping. TrapNet employs graph community detection techniques for malware fingerprinting and family attribution based on static analysis, as follows: (1) TrapNet detects packed binaries and unpacks them using known generic packer tools. (2) From each malware sample, it generates a digest that captures the underlying semantics. Since the digest must be dense, efficient, and suitable for similarity checking, we designed FloatHash (FH), a novel numerical fuzzy hashing technique that produces a short real-valued vector summarizing the underlying assembly items and their order. FH is based on applying Principal Component Analysis (PCA) to ordered assembly items (e.g., opcodes, function calls) extracted from the malware's assembly code. (3) Representing malware with short numerical vectors enables high-performance, large-scale similarity computation, which allows TrapNet to build a malware similarity network. (4) Finally, TrapNet employs state-of-the-art community detection algorithms to identify dense communities, which represent groups of malware with similar semantics. Our extensive evaluation of TrapNet demonstrates its effectiveness in terms of the coverage and purity of the detected communities, while also highlighting its runtime efficiency, which outperforms other state-of-the-art solutions.

Canonical correlation analysis (CCA) is a technique for finding correlated sets of features between two datasets. In this paper, we propose a novel extension of CCA to the online, streaming data setting: Sliding Window Informative Canonical Correlation Analysis (SWICCA). Our method uses a streaming principal component analysis (PCA) algorithm as a backend and uses these outputs combined with a small sliding window of samples to estimate the CCA components in real time. We motivate and describe our algorithm, provide numerical simulations to characterize its performance, and provide a theoretical performance guarantee. The SWICCA method is applicable and scalable to extremely high dimensions, and we provide a real-data example that demonstrates this capability.

Hyperparameter tuning is a key component of machine learning procedures, but when validation data contain sensitive user information, search mechanisms can leak private information through the selected configuration. Existing differentially private hyperparameter tuning methods often rely on near-random search, while prior differentially private Bayesian optimization approaches are typically global and, therefore, scale poorly with the hyperparameter dimensionality. We study differentially private hyperparameter tuning using local Bayesian optimization, focusing on settings where the validation objective is available only through noisy black box evaluations and gradients are unavailable or impractical to compute. We introduce DP-GIBO, a differentially private local Bayesian optimization framework that privately approximates gradients using a Gaussian Process surrogate. Under suitable conditions, we prove that DP-GIBO converges to a locally optimal hyperparameter configuration up to a privacy-dependent error, with dimensional dependence that is polynomial rather than exponential.Empirically, we show that DP-GIBO provides scalable private hyperparameter tuning across multiple tasks, substantially outperforming non-private random search and global Bayesian optimization baselines in moderate-to-high-dimensional hyperparameter spaces.

We study the estimation of distributional parameters when samples are shown only if they fall in some unknown set $S \subseteq \mathbb{R}^d$. Kontonis, Tzamos, and Zampetakis (FOCS'19) gave a $d^{\mathrm{poly}(1/\varepsilon)}$ time algorithm for finding $\varepsilon$-accurate parameters for the special case of Gaussian distributions with diagonal covariance matrix. Recently, Diakonikolas, Kane, Pittas, and Zarifis (COLT'24) showed that this exponential dependence on $1/\varepsilon$ is necessary even when $S$ belongs to some well-behaved classes. These works leave the following open problems which we address in this work: Can we estimate the parameters of any Gaussian or even extend beyond Gaussians? Can we design $\mathrm{poly}(d/\varepsilon)$ time algorithms when $S$ is a simple set such as a halfspace? We make progress on both of these questions by providing the following results: 1. Toward the first question, we give a $d^{\mathrm{poly}(\ell/\varepsilon)}$ time algorithm for any exponential family that satisfies some structural assumptions and any unknown set $S$ that is $\varepsilon$-approximable by degree-$\ell$ polynomials. This result has two important applications: 1a) The first algorithm for estimating arbitrary Gaussian distributions from samples truncated to an unknown $S$; and 1b) The first algorithm for linear regression with unknown truncation and Gaussian features. 2. To address the second question, we provide an algorithm with runtime $\mathrm{poly}(d/\varepsilon)$ that works for a set of exponential families (containing all Gaussians) when $S$ is a halfspace or an axis-aligned rectangle. Along the way, we develop tools that may be of independent interest, including, a reduction from PAC learning with positive and unlabeled samples to PAC learning with positive and negative samples that is robust to certain covariate shifts.

Symmetry is ubiquitous in many real-world phenomena and tasks, such as physics, images, and molecular simulations. Empirical studies have demonstrated that incorporating symmetries into generative models can provide better generalization and sampling efficiency when the underlying data distribution has group symmetry. In this work, we provide the first theoretical analysis and guarantees of score-based generative models (SGMs) for learning distributions that are invariant with respect to some group symmetry and offer the first quantitative comparison between data augmentation and adding equivariant inductive bias. First, building on recent works on the Wasserstein-1 ($\mathbf{d}_1$) guarantees of SGMs and empirical estimations of probability divergences under group symmetry, we provide an improved $\mathbf{d}_1$ generalization bound when the data distribution is group-invariant. Second, we describe the inductive bias of equivariant SGMs using Hamilton-Jacobi-Bellman theory, and rigorously demonstrate that one can learn the score of a symmetrized distribution using equivariant vector fields without data augmentations through the analysis of the optimality and equivalence of score-matching objectives. This also provides practical guidance that one does not have to augment the dataset as long as the vector field or the neural network parametrization is equivariant. Moreover, we quantify the impact of not incorporating equivariant structure into the score parametrization, by showing that non-equivariant vector fields can yield worse generalization bounds. This can be viewed as a type of model-form error that describes the missing structure of non-equivariant vector fields. Numerical simulations corroborate our analysis and highlight that data augmentations cannot replace the role of equivariant vector fields.

The combination of machine learning (ML) and sparsity-promoting techniques is enabling direct extraction of governing equations from data, revolutionizing computational modeling in diverse fields of science and engineering. The discovered dynamical models could be used to address challenges in climate science, neuroscience, ecology, finance, epidemiology, and beyond. However, most existing sparse identification methods for discovering dynamical systems treat the whole system as one without considering the interactions between subsystems. As a result, such models are not able to capture small changes in the emergent system behavior. To address this issue, we developed a new method called Sparse Identification of Nonlinear Dynamical Systems from Graph-structured data (SINDyG), which incorporates the network structure into sparse regression to identify model parameters that explain the underlying network dynamics. We tested our proposed method using several case studies of neuronal dynamics, where we modeled the macroscopic oscillation of a population of neurons using the extended Stuart-Landau (SL) equation and utilize the SINDyG method to identify the underlying nonlinear dynamics. Our extensive computational experiments validate the improved accuracy and simplicity of discovered network dynamics when compared to the original SINDy approach. The proposed graph-informed penalty can be easily integrated with other symbolic regression algorithms, enhancing model interpretability and performance by incorporating network structure into the regression process.

We give a unified explanation of the geometric and algebraic properties of two well-known maps, one from permutations to triangulations, and another from permutations to subsets. Furthermore we give a broad generalization of the maps. Specifically, for any lattice congruence of the weak order on a Coxeter group we construct a complete fan of convex cones with strong properties relative to the corresponding lattice quotient of the weak order. We show that if a family of lattice congruences on the symmetric groups satisfies certain compatibility conditions then the family defines a sub Hopf algebra of the Malvenuto-Reutenauer Hopf algebra of permutations. Such a sub Hopf algebra has a basis which is described by a type of pattern avoidance. Applying these results, we build the Malvenuto-Reutenauer algebra as the limit of an infinite sequence of smaller algebras, where the second algebra in the sequence is the Hopf algebra of non-commutative symmetric functions. We also associate both a fan and a Hopf algebra to a set of permutations which appears to be equinumerous with the Baxter permutations.

Newman, Schneider and Shalev defined the entropy of a graded associative algebra A as H(A) = \limsup_{n \to \infty} \sqrt[n]{a_n}, where a_n is the vector space dimension of the n'th homogeneous component. When A is the homogeneous quotient of a finitely generated free associative algebra, they showed that H(A) \le \sqrt{a_2}. Using some results of Friedland on the maximal spectral radius of 0-1 matrices with a prescribed number of ones, we improve on this bound.

We study the average-case hardness of establishing that a graph does not have a large clique in both proof and communication complexity. We show exponential lower bounds on the length of cutting planes and bounded-depth resolution over parities refutations of the binary encoding of clique formulas on randomly sampled dense graphs. Moreover, we show that the randomized communication complexity of finding a falsified clause in these formulas is polynomial.

On November 12th, 2025 the LIGO--Virgo--KAGRA (LVK) collaboration reported gravitational waves (GWs) from a compact object merger candidate (S251112cm) with at least one sub-solar mass component. Using the Dark Energy Camera (DECam), the Fraunhofer Telescope at Wendelstein Observatory (FTW), and the Zwicky Transient Facility (ZTF), we surveyed $56\%$ of the GW localization region beginning $2.4$~hours after the GW alert. We find no kilonova (KN) counterpart, and use radiative-transfer models to rule out $42\%$ (ZTF), $68\%$ (DECam), and $92\%$ (FTW) of the KN models as possible emission from this GW candidate. Within the recently proposed disk-fragmentation (``superkilonova'') model for generating sub-solar mass neutron star mergers from stellar core-collapse, the delay between the supernova explosion time and the GW merger time is estimated to be less than a few days. Searching this time window prior to the GW event, we identify and spectroscopically classify a IIb supernova (SN~2025adtq), with a spatial association odds ratio of $\log_{10}\mathcal{I} \approx 4.8$, a chance coincidence probability of ${\sim}2$--$9\%$, and an estimated explosion time ${\sim}2$ days prior to S251112cm. SN~2025adtq is the second Type~IIb supernova found in spatial and temporal coincidence with a sub-solar mass GW candidate, following the previously reported S250818k/SN~2025ulz association; jointly, we measure an odds ratio that favors the association hypothesis over the null, however, when conditioned on finding a coincident supernova by chance, the odds ratio disfavors association. Together, these results provide suggestive but inconclusive evidence for the superkilonova formation channel.

Let $K \subset \mathbb{R}^n$ be a centered convex body of volume one. We prove that there exist absolute constants $c,C > 0$ and an orthonormal set of vectors $Θ\subset S^{n-1}$ with size $\left|Θ\right| \ge 9n/10$ such that, if $X$ is a random vector uniformly distributed on $K$, then for all $θ\in Θ$ one has \[ c\cdot \sqrt{p}\,\left(\mathbb{E} \left|\left\langle X,θ\right\rangle\right|^2\right)^{1/2} \le \left(\mathbb{E} \left|\left\langle X,θ\right\rangle\right|^p\right)^{1/p} \le C\cdot \sqrt{p}\,\left(\mathbb{E} \left|\left\langle X,θ\right\rangle\right|^2\right)^{1/2}, \] where the upper estimate holds for all $p \ge 1$ while the lower bound only holds for $1 \le p \le n$.

The Spherical Fourier-Bessel (SFB) basis, in separating the angular and radial modes of the power spectrum, permits a targeted identification and mitigation of systematics in clustering surveys while retaining more cosmological signal than traditional bases. We demonstrate this principle on the eBOSS DR16 LRG and QSO samples, identifying modes which may be contaminated by systematics. Our initial inference on the LRG sample yields an fNL value consistent with zero, while the QSO value is in slight tension with zero. Using the SFB basis, we vary the selection of angular and radial modes to search for inconsistencies in the inferred value of fNL, an indicator of underlying systematics. In the QSO sample, we find evidence (p < 0.005 compared to the same cuts on EZMocks) of a systematic afflicting large physical scales, which is consistent with residual stellar contamination; we also find evidence (p < 0.05) for an unknown systematic in the QSO and LRG samples at the approximate angular plate and imaging scale of eBOSS.

Solid-state processors require control layers whose errors are legible to quantum-error-correction decoders. We show that crystallographic symmetry can provide such a layer in strain-active Lambda manifolds. When the projected strain tensor and Lambda-transition operators share a multiplicity-one two-dimensional irreducible representation, symmetry fixes the linear strain interaction to a scalar dot product. Two phase-locked mechanical modes synthesize a circular strain field, enabling complex phononic Lambda-leg control without local microwave near fields. On this manifold we construct a superadiabatic echo-lune holonomic gate using Lambda-leg control and a resonant double-quantum counterdiabatic tone. Rotating-frame simulations of a nitrogen-vacancy center give 99.88% conditional average fidelity in 1.833 microseconds, or 99.40% when leakage is counted as error. A resonant gigahertz high-overtone bulk acoustic resonator analysis translates the Hamiltonian into Rabi-rate, linewidth, and envelope-tracking requirements. The bright-state structure organizes noise: A2-sector perturbations are parity-filtered into an optically distinguishable auxiliary state, whereas transverse E-sector faults are echo suppressed and retained as a decoder stress axis. The extracted channel has 0.47% erasure probability and 0.168% residual Z error. In XZZX code-capacity simulations, this biased-erasure model yields a nominal 64% fit-extrapolated data-qubit reduction relative to an unstructured Rabi baseline. Repeated-round detector-model diagnostics preserve the nominal distance-9 proxy and identify missed erasures, transverse floors, leakage/flag timing, and strong crosstalk as validation limits. Extensions to orbital Lambda systems and bright-projector phonon-bus diagnostics identify crystallographic symmetry as a principle for co-designing phononic actuation, leakage, noise bias, and quantum decoding.

Preserving the admissible set of the ideal magnetohydrodynamics (MHD) equations is important not only for producing physically meaningful numerical solutions, but more importantly for achieving robust computations. In this paper, we develop an optimization-based limiter to enforce admissibility while preserving global conservation and accuracy. For an easy and efficient projection, we decompose the admissible set into slices parameterized by the magnetic energy, so that the MHD projection reduces to a one-dimensional minimization, which can be solved efficiently by the Brent method. The splitting method can be used to efficiently solve the global minimization problem of the optimization-based limiter, which can be used to enforce cell average admissibility in discontinuous Galerkin (DG) schemes, and pointwise admissibility can be further enforced by the Zhang-Shu positivity-preserving limiter. We apply the limiter to high-order DG schemes and present numerical results for a few representative MHD problems.

The redshifted 21\,cm line is an emerging tool in observational cosmology that can serve as a direct probe of the intergalactic medium throughout the cosmic timeline. However, the observation of the cosmological 21\,cm signal from early epochs is extremely challenging in practice, regardless of the scale of interest and redshift. The presence of bright astrophysical foregrounds and residual systematic errors along the line of sight poses challenges for its detection. Machine-learning-based Gaussian process regression\,(ML-GPR) has proven to be the most effective strategy for signal separation in LOFAR and NenuFAR observations to measure the 21\,cm signal power spectrum from the Cosmic Dawn\,(CD) and Epoch of Reionization\,(EoR). In this work, we extend this framework to synthetic CD/EoR SKA1-Low observations to assess its robustness in mitigating residual foregrounds against instrumental and environmental systematic effects. We use our developed end-to-end realistic simulation pipeline (\textsc{21cmE2E}) for SKA-Low observations. Our 4-hour tracking simulation includes extragalactic point sources, the AA* telescope configuration, primary beam response, and error models. The modelled errors incorporate residual antenna-based gain calibration errors, residual ionospheric phase errors, partial de-mixing of the out-of-field sources, and instrumental noise for 1000\,hours of deep integration time. We compare different Bayesian GPR frameworks to assess their ability to suppress residual foreground contamination while minimizing signal loss and providing reliable uncertainty estimates. Our analysis demonstrates that the 21\,cm signal can robustly recover within the $2σ$ credible interval for almost all k-modes over the range of $0.06 \leq k \leq 1.0$~h Mpc$^{-1}$.

We study deterministic online algorithms for the problem of chasing sets of cardinality at most $k$ in a metric space, also known as metrical service systems and equivalent to width-$k$ layered graph traversal. We resolve the 30-year-old gap of $Ω(2^k)\cap O(k2^k)$ on the competitive ratio of this problem by giving an $O(2^k)$-competitive deterministic algorithm. This bound is optimal even among randomized algorithms against adaptive adversaries. We also (slightly) improve the deterministic lower bound to $D_k$, defined recursively by $D_1=1$ and $D_{k+1}=2D_k+\sqrt{8+8D_k}+3$, which we conjecture to be exactly tight. For $k=3$, we provide a matching upper bound of $D_3$. Our results imply slightly improved upper and lower bounds for distributed asynchronous collective tree exploration and for the $k$-taxi problem, respectively. Our algorithm generalizes the classical doubling strategy, previously known to be optimal for $k=2$. The previous best bound for general $k$ was achieved by the generalized work function algorithm (WFA), and was known to be tight for WFA. Our improved bound therefore implies that WFA is sub-optimal for chasing small sets.

We study the enumeration of spinal tree-child phylogenetic networks, a rigid family of tree-child networks in which all internal vertices lie on a single root--to--leaf path. We provide two complementary combinatorial frameworks. First, we introduce a word model: unlabeled spinal networks correspond to a suitable class of restricted words with fixed multiplicities, taken modulo a simple relabeling equivalence, which yields an explicit closed enumeration. Second, we develop a symbolic-method approach based on a marked version of trees that admits a clean recursive specification; its boxed-product translation leads to a solvable bivariate generating function and a direct derivation of the coefficients.

Software developers use metrics to evaluate code quality and productivity, but these practices are still rare in programming education. This project bridges the gap by collecting real-time learning analytics from individual student and whole-class code development logs. This granular, quantitative data provides educators with qualitative insights into the learning process. It allows them to evaluate student comprehension, identify common challenges, and critically assess whether the allocated time for exercises and algorithms is sufficient for mastery. Unlike traditional Learning Management Systems, we propose a novel approach: a plugin for a widely used code editor that captures granular interactions during programming and documentation. The resulting dataset logs coding behaviors, errors, and progress, enabling evidence-based analysis of learning patterns and educational benchmarking. By structuring this real-time programming trail, we support research on teaching methodologies, learner challenges, and skill acquisition. Quantitative metrics complement qualitative assessment by evaluating code, exercise progress, and timestamp logs. Our goal is to provide an open-access database for educators and researchers, fostering data-driven insights to enhance instruction and personalize learning experiences. This work aligns industrial best practices with pedagogical innovation, advancing measurable, empirical approaches to programming education.

Ozone (O3) is a triatomic molecule of central importance in the chemistry and physics of the Earth's and other planetary atmospheres. Beyond its environmental significance, a detailed understanding of the electronic structure and ionization dynamics of ozone is essential for modeling atmospheric, ionospheric, and astrochemical processes. In the present work, we substantially extend the experimental and theoretical characterization of ozone into the regime of valence double photoionization. Using HeII-alpha, HeII-beta, and higher-energy vacuum ultraviolet radiation in combination with a versatile multiple charged-particle correlation detection technique, we report the first single-photon valence double ionization electron spectrum of O3. To interpret the experimental observations, we mapped the lowest potential energy surfaces of dicationic ozone employing post-Hartree-Fock multi-configurational-interaction methods, and computed with high accuracy the energetics of the relevant dissociation channels. Our results demonstrate that dissociative double ionization of ozone produces electronically excited cationic atomic oxygen fragments in addition to the ground-state dissociation pathway, revealing a richer fragmentation dynamics than hitherto recognized.

We consider a high quantile homogeneity test to determine whether a certain set of explanatory variables has homogeneous effects on different high quantiles of the response variable in the tail. To accommodate for situations under both the null and the alternative, the auxiliary process in this case may no longer be treated as stationary, and the problem requires a joint analysis of both homoscedastic and heteroskedastic high quantiles. For this, we develop a novel Bahadur representation result in the high quantile setting for a general class of tail dependent time series under potential heteroskedasticity, which can be of interest by its own. In particular, the Bahadur representation provides a foundation for reducing problems regarding nonlinear high quantile regression estimators to those regarding suitably constructed linear forms with an explicit error bound and can be transformative and useful in many statistical problems. We in the current article apply it to guide the development of a generative high quantile homogeneity test, which is then illustrated through applications to both synthetic and real data.

State-transition models are essential across epidemiology and ecology, but statistical inference remains challenging owing to high-dimensional latent state spaces, temporal dependence, and intractable likelihood functions. Bayesian inference via Markov Chain Monte Carlo (MCMC) enables joint estimation of model parameters and missing event times through data augmentation, but Metropolis-within-Gibbs (MWG) schemes that combine multiple specialised kernels are notoriously difficult to implement. Current probabilistic programming frameworks face a trade-off: automation sacrifices extensibility, whilst flexibility demands substantial implementation overhead. This divide has created a software landscape characterised by tightly coupled, model-specific implementations that resist reuse and extension. We introduce gemlib.mcmc, an MCMC module designed to bridge methodological and applied communities through principled, composable kernel abstractions. The framework employs writer monads from category theory to formalise kernel composition, enabling seamless integration of parameter-estimation and data-augmentation kernels without manual state management. Built on JAX and TensorFlow Probability for high-performance computation, gemlib.mcmc provides an ergonomic interface -- leveraging Python's right-shift operator for intuitive kernel chaining -- whilst maintaining statistical rigour and transparency. Developers can extend the library by implementing only two methods; composition and hardware acceleration are automated. We demonstrate the framework through parameter inference on partially observed epidemic models, showing how complex inference algorithms can be expressed concisely and reused across applications. By reducing implementation burden we provide access to sophisticated MCMC methods and enable applied researchers to employ state-of-the-art algorithms without reimplementation overhead.

The overlap gap property (OGP) is a statement about the geometry of near-optimal solutions. Exhibiting OGP implies failure of a class of local algorithms; and has been observed to coincide with conjectured algorithmic limits in problems with statistical computational gap. We consider the Stochastic Block Model (SBM), where the graph has a planted partition with $k$ equal-size blocks which form the `communities', and where, for parameters $p>q$, vertices within the same community connect with probability $p$, while vertices in different communities connect with probability $q$, independently across pairs of vertices. Modularity--based clustering algorithms have become ubiquitous in applications. This article studies theoretical limits of local algorithms based on the modularity score on the SBM. We establish that modularity exhibits OGP on the SBM. This rules out a class of local algorithms based on modularity for recovery in the SBM, and shows slow mixing time for a related Markov Chain. Theoretically this is one of the few instances where OGP has been established for a `planted' model, as most such analyses to date consider the `null' model. As part of our analysis, we extend a result by Bickel and Chen 2009, who established that with high probability, the modularity optimal partition of SBM is $o(n)$ local moves away from the planted partition, where $n$ is the graph size. We show that, with high probability, any partition with modularity score sufficiently near the optimal value is close to the planted partition.