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We consider Bernoulli bond percolation with fixed retention parameter $p$ on the random recursive tree, coupled through the natural growth process. We prove that the root cluster has a strictly positive probability of remaining a largest cluster at every time. Equivalently, in the associated Simon-type Chinese restaurant process, the first table has a positive probability of remaining a largest table forever. We further show that two associated quantities, the limit as $n\to\infty$ of the probability that the root cluster is a largest cluster at time $n$ and the probability that it remains a largest cluster for all times, are strictly increasing and continuous functions of $p$ on $(0,1]$, and both tend to zero as $p\downarrow0$.

Aligned large language models (LLMs) remain vulnerable to jailbreak attacks. Recent mechanistic studies have identified latent features and representation shifts associated with jailbreak success, but they leave a more fundamental question open: why do aligned LLMs remain jailbreakable, and what structural vulnerabilities in the model make this possible? We study this question through a continuous input-transformation view. Our theoretical finding is that aligned models can still exhibit Refusal-Escape Directions (RED): local perturbation directions around a harmful input that shift the model's behavior from refusal to answering while preserving the model's harmful-semantics interpretation. From this perspective, a jailbreak is not only a successful discrete prompt construction, but can also be understood as a refusal-to-answer behavior transition induced by continuously perturbing a harmful input along RED. We then prove that RED can be exactly decomposed into contributions from operator-level sources across the model's operator structure, and identify normalization, residual-wiring, and terminal sources as analytically constrained operator-level sources. To eliminate RED, the sha

In non-Hermitian physics, high-order exceptional points(HOEPs) with eigenvalues and eigenvectors coalesce are known for their enhanced sensitivity to perturbations. Typically, they exhibit eigenvalue splitting that scales as ε^(1/n), which is referred to as the generic response. However, under certain conditions, a nongeneric response of HOEPs occurs where the splitting follows a lower order ε^(1/m) (m<n). A nongeneric response of HOEPs with a lower order splitting lead to the remaining EPs. While the presence of these remaining EPs is acknowledged, a thorough elucidation of their fundamental properties has yet to be achieved. In this work, we demonstrate those unsplit eigenvalue points must constitute remaining EPs in a perturbed n-orders HOEPs system. Combining graph theory and topological analysis, the number and splitting order of the remaining EPs is studied. This framework not only resolves a fundamental challenge in HOEPs but also paves the way for exploiting remaining EPs in applications such as anisotropic sensing and the design of Dirac exceptional points.

This survey-style note reviews constructive versions of the Peter--Weyl theorem in the Bishop--Coquand--Spitters line. Its main purpose is to clarify which parts of the classical Peter--Weyl package admit constructive reformulations, which parts survive only in weaker or reorganized form, and which questions still appear to remain open. The term ``constructive'' is used here primarily in the Bishop-style sense, together with the related locale-theoretic and formal-topological developments that occur in the work of Coquand and Spitters. We review the constructive compact-group results of Coquand and Spitters, the later role of almost periodic functions and compact completions, and the interaction with constructive Gelfand representation and locale-theoretic compactness. The guiding theme is that the constructive theory exists, but it is often most naturally expressed not as a literal transcription of the classical theorem in terms of irreducible decompositions alone, but rather through finite-rank approximation, characters, and compactifications attached to functions or groups. For orientation and comparison, we also include an appendix giving a standard classical form of the Peter-

With the objective of characterizing the stationary behavior of the scaling limit for shortest remaining processing time (SRPT) queues with a heavy-tailed processing time distribution, as obtained in Banerjee, Budhiraja, and Puha (BBP, 2022), we study reflecting coupled Brownian motions (RCBM) $(W_t(a), a, t \geq 0)$. These RCBM arise by regulating coupled Brownian motions (CBM) $(χ_t(a), a,t \geq 0)$ to remain nonnegative. Here, for $t\geq 0$, $χ_t(0)=0$ and $χ_t(a):=w(a)+σB_t-μ(a)t$ for $a>0$, $w(\cdot)$ is a suitable initial condition, $σ$ is a positive constant, $B$ is a standard Brownian motion, and $μ(\cdot)$ is an unbounded, positive, strictly decreasing drift function. In the context of the BBP (2022) scaling limit, the drift function is determined by the model parameters, and, for each $a\geq 0$, $W_{\cdot}(a)$ represents the scaling limit of the amount of work in the system of size $a$ or less. Thus, for the BBP (2022) scaling limit, the time $t$ values of the RCBM describe the random distribution of the size of the remaining work in the system at time $t$. Our principal results characterize the stationary distribution of the RCBM in terms of a maximum process $M_*(\cd

Retrieval-augmented generation (RAG) allows large language models to access external and private corpora for factual, domain-specific responses. Modern RAG pipelines use hierarchical navigable small world (HNSW) vector databases for efficient similarity search. When a user requests data deletion, the systems typically only mark the record as deleted, leaving the embedding on disk physically unchanged. This soft-delete operation raises compliance concerns under data-erasure and retention requirements such as GDPR Article 17 and HIPAA. Analysis on three HNSW implementations confirms that deleted vectors remain physically recoverable by accessing the raw index files at the storage layer, bypassing API access. Using the Vec2Text inversion model without domain-specific fine-tuning, we show this vulnerability on multiple real-world datasets and data modalities. On Wikipedia biographical living persons dataset (BLP), we successfully recover 25.5% of exact person names and 46.4% of geographic locations (ROUGE-L 0.185). Recovery reaches 100% for both patient age and gender markers (ROUGE-L 0.290) on highly structured, sensitive data (NIH Synthea dataset). On soft-deleted image embeddings, w

Chebyshev points are distinguished in polynomial interpolation by the logarithmic growth of their Lebesgue constants. This paper asks a simple question: how much can Chebyshev points be perturbed before they cease to behave like Chebyshev points? We study perturbed Chebyshev--Lobatto nodes $x_j=\cos(jπ/n+\varepsilon_j)$, with angular perturbations $|\varepsilon_j|\leq σ_n$. The study is motivated by numerical experiments showing a broad stable region when the mesh fraction $nσ_n$ is small and rapid amplification for larger perturbations; the observed transition region is consistent with the curve $nσ_n\asymp(\log n)^{-1}$. The main result is a deterministic worst-case stability estimate: if $nσ_n(\log n+1)$ is bounded by a sufficiently small constant, then the Lebesgue constant remains logarithmic. The proof uses the cosine parametrization and Bernstein's inequality for trigonometric polynomials, thereby exploiting the angular geometry of the Chebyshev--Lobatto grid rather than a Markov inequality in the physical variable. We also give a worst-case obstruction at the angular mesh scale, showing that perturbations of order $1/n$ cannot be allowed uniformly. Consequences are derived

Predicting mortality-related outcomes from images offers the prospect of accessible, noninvasive, and scalable health screening. We present a method that leverages pretrained vision transformer foundation models to estimate remaining lifespan from facial and whole-body images, alongside robust uncertainty quantification. We show that predictive uncertainty varies systematically with the true remaining lifespan, and that this uncertainty can be effectively modeled by learning a Gaussian distribution for each sample. Our approach achieves state-of-the-art mean absolute error (MAE) of 7.41 years on an established dataset, and further achieves 4.91 and 4.99 years MAE on two new, higher-quality datasets curated and published in this work. Importantly, our models provide calibrated uncertainty estimates, as demonstrated by a bucketed expected calibration error of 0.82 years on the Faces Dataset. While not intended for clinical deployment, these results highlight the potential of extracting medically relevant signals from images. We make all code and datasets available to facilitate further research.

The battery state of health (SOH) based on capacity fade and resistance increase is not sufficient for predicting Remaining Useful life (RUL). The electrochemical community blames the path-dependency of the battery degradation mechanisms for our inability to forecast the degradation. The control community knows that the path-dependency is addressed by full state estimation. We show that even the electrode-specific SOH (eSOH) estimation is not enough to fully define the degradation states by simulating infinite possible degradation trajectories and remaining useful lives (RUL) from a unique eSOH. We finally define the deepSOH states that capture the individual contributions of all the common degradation mechanisms, namely, SEI, plating, and mechanical fracture to the loss of lithium inventory. We show that the addition of cell expansion measurement may allow us to estimate the deepSOH and predict the remaining useful life.

Signal processing makes extensive use of point estimators and accompanying error bounds. These work well up until the likelihood function has two or more high peaks. When it is important for an estimator to remain reliable, it becomes necessary to consider alternatives, such as set estimators. An obvious first choice might be confidence intervals or confidence regions, but there can be difficulties in computing and interpreting them (and sometimes they might still be blind to multiple peaks in the likelihood). Bayesians seize on this to argue for replacing confidence regions with credible regions. Yet Bayesian statistics require a prior, which is not always a natural part of the problem formulation. This paper demonstrates how a re-interpretation of the prior as a weighting function makes an otherwise Bayesian estimator meaningful in the frequentist context. The weighting function interpretation also serves as a reminder that an estimator should always be designed in the context of its intended application; unlike a prior which ostensibly depends on prior knowledge, a weighting function depends on the intended application. This paper uses the time-of-arrival (TOA) problem to illust

Lithium-ion batteries are widely used in various applications, including portable electronic devices, electric vehicles, and renewable energy storage systems. Accurately estimating the remaining useful life of these batteries is crucial for ensuring their optimal performance, preventing unexpected failures, and reducing maintenance costs. In this paper, we present a comprehensive review of the existing approaches for estimating the remaining useful life of lithium-ion batteries, including data-driven methods, physics-based models, and hybrid approaches. We also propose a novel approach based on machine learning techniques for accurately predicting the remaining useful life of lithium-ion batteries. Our approach utilizes various battery performance parameters, including voltage, current, and temperature, to train a predictive model that can accurately estimate the remaining useful life of the battery. We evaluate the performance of our approach on a dataset of lithium-ion battery cycles and compare it with other state-of-the-art methods. The results demonstrate the effectiveness of our proposed approach in accurately estimating the remaining useful life of lithium-ion batteries.

Machine unlearning (MU) has emerged to enhance the privacy and trustworthiness of deep neural networks. Approximate MU is a practical method for large-scale models. Our investigation into approximate MU starts with identifying the steepest descent direction, minimizing the output Kullback-Leibler divergence to exact MU inside a parameters' neighborhood. This probed direction decomposes into three components: weighted forgetting gradient ascent, fine-tuning retaining gradient descent, and a weight saliency matrix. Such decomposition derived from Euclidean metric encompasses most existing gradient-based MU methods. Nevertheless, adhering to Euclidean space may result in sub-optimal iterative trajectories due to the overlooked geometric structure of the output probability space. We suggest embedding the unlearning update into a manifold rendered by the remaining geometry, incorporating second-order Hessian from the remaining data. It helps prevent effective unlearning from interfering with the retained performance. However, computing the second-order Hessian for large-scale models is intractable. To efficiently leverage the benefits of Hessian modulation, we propose a fast-slow parame

Machine unlearning (MU) aims to remove the influence of specific training samples from a well-trained model, a task of growing importance due to the ``right to be forgotten.'' The unlearned model should approach the retrained model, where forgetting data do not contribute to the training process. Therefore, unlearning should withdraw their contribution from the pre-trained model. However, quantifying and disentangling sample's contribution to overall learning process is highly challenging, leading most existing MU approaches to adopt other heuristic strategies such as random labeling or knowledge distillation. These operations inevitably degrade model utility, requiring additional maintenance with remaining data. To advance MU towards better utility and efficiency for practical deployment, we seek to approximate sample contribution with only the pre-trained model. We theoretically and empirically reveal that sample's contribution during training manifests in the learned model's increased sensitivity to it. In light of this, we propose MU-Mis (Machine Unlearning by Minimizing input sensitivity), which directly suppresses the contribution of forgetting data. This straightforward supp

Background. Women bring unique problem-solving skills to software development, often favoring a holistic approach and attention to detail. In software testing, precision and attention to detail are essential as professionals explore system functionalities to identify defects. Recognizing the alignment between these skills and women's strengths can derive strategies for enhancing diversity in software engineering. Goal. This study investigates the motivations behind women choosing careers in software testing, aiming to provide insights into their reasons for entering and remaining in the field. Method. This study used a cross-sectional survey methodology following established software engineering guidelines, collecting data from women in software testing to explore their motivations, experiences, and perspectives. Findings. The findings reveal that women enter software testing due to increased entry-level job opportunities, work-life balance, and even fewer gender stereotypes. Their motivations to stay include the impact of delivering high-quality software, continuous learning opportunities, and the challenges the activities bring to them. However, inclusiveness and career developme

The Tolman-Ehrenfest criterion of thermal equilibrium for a static fluid in a static spacetime is generalized to stationary heat conduction, in the approximation in which backreaction is negligible. Applying this generalized criterion to the Hawking radiation in the Schwarzschild-de Sitter geometry shows that the two horizons (which act as thermostats) remain in thermal equilibrium. The temperature of the radiation fluid interpolates between the temperatures at the horizons, with a static analytic profile that is given explicitly.

We study age-structured branching models with reproduction law depending on the remaining lifetime of the parent. The lifespan of an individual is decided at its birth and its remaining lifetime decreases at the unit speed. The models without or with immigration are constructed as measure-valued processes by pathwise unique solutions of stochastic equations driven by time-space Poisson random measures. In the subcritical branching case, we give a sufficient condition for the ergodicity of the process with immigration. Two large number laws and a central limit theorem of the occupation times are proved.

Structured electronic health records (EHR) are essential for clinical prediction. While count-based learners continue to perform strongly on such data, no benchmarking has directly compared them against more recent mixture-of-agents LLM pipelines, which have been reported to outperform single LLMs in various NLP tasks. In this study, we evaluated three categories of methodologies for EHR prediction using the EHRSHOT dataset: count-based models built from ontology roll-ups with two time bins, based on LightGBM and the tabular foundation model TabPFN; a pretrained sequential transformer (CLMBR); and a mixture-of-agents pipeline that converts tabular histories to natural-language summaries followed by a text classifier. We assessed eight outcomes using the EHRSHOT dataset. Across the eight evaluation tasks, head-to-head wins were largely split between the count-based and the mixture-of-agents methods. Given their simplicity and interpretability, count-based models remain a strong candidate for structured EHR benchmarking. The source code is available at: https://github.com/cristea-lab/Structured_EHR_Benchmark.

We study supercritical age-structured branching models starting from a single particle with a random lifetime, where the reproduction law depends on the remaining lifetime of the parent. The lifespan of an individual is decided at its birth and its remaining lifetime decreases at the unit speed. A necessary and sufficient condition is provided for the convergence of the Malthusian normalized random measures. The Malthusian type limit theory in a functional form can be strengthened to hold with probability one under some ``$L\log L$'' conditions. We further prove a central limit theory with a random normalization factor.

The face is a rich source of information that can be utilized to infer a person's biological age, sex, phenotype, genetic defects, and health status. All of these factors are relevant for predicting an individual's remaining lifespan. In this study, we collected a dataset of over 24,000 images (from Wikidata/Wikipedia) of individuals who died of natural causes, along with the number of years between when the image was taken and when the person passed away. We made this dataset publicly available. We fine-tuned multiple Convolutional Neural Network (CNN) models on this data, at best achieving a mean absolute error of 8.3 years in the validation data using VGGFace. However, the model's performance diminishes when the person was younger at the time of the image. To demonstrate the potential applications of our remaining lifespan model, we present examples of using it to estimate the average loss of life (in years) due to the COVID-19 pandemic and to predict the increase in life expectancy that might result from a health intervention such as weight loss. Additionally, we discuss the ethical considerations associated with such models.